When the new Liaison Committee finally meets today, its choice of chair will be one of the highest-profile and potentially critical decisions taken by any select committee in this Parliament.

The previous incumbent, then-Treasury Committee chair Andrew Tyrie, stood down at the general election after filling the role since 2015.

Liaison Committee chairs are traditionally members of a/the governing party, experienced select committee members or (since 2010 serving) chairs, and male.

This year, of the 15 Conservative Liaison Committee members whose names are known, just five have been chairs of their committees continuously since before the 2015 election: Bill Cash (European Scrutiny, 2010); David T. C. Davies (Welsh Affairs, 2010); Bernard Jenkin (Public Administration and Constitutional Affairs, 2010); Charles Walker (Procedure, 2012); and Sarah Wollaston (Health, 2014).

Of these, all but Sarah Wollaston could before the end of this Parliament fall foul of the current eight-year term limit for select committee chairs and be obliged to leave the Liaison Committee - although, as the Speaker confirmed in July, this limit may be amended or repealed before it first has effect in June 2018.

Of course, several of the other Conservative Liaison Committee members have considerable select committee experience accumulated before becoming chairs. And a new factor this year will be whether and how candidates’ Brexit positions affect the Liaison Committee’s choice. According to the BBC, of the five pre-2015 Conservative chairs on the Liaison Committee, four backed ‘Leave’; Sarah Wollaston famously switched from ‘Leave’ to ‘Remain’ with two weeks to go.

On Brexit, the Liaison Committee has a potentially critical role.

This is because Brexit is an all-of-government project requiring political choices between departments, sectors and priorities that can only be made by the Prime Minister, and administrative effectiveness that can only be enforced through the centre of government under her control.

The Liaison Committee’s most high-profile function is the taking of public evidence from the Prime Minister three times a year, for far longer each time (90 minutes) than is offered by PMQs or any broadcast interview. This makes the Liaison Committee the only body that might achieve in-depth scrutiny of the Prime Minister’s Brexit actions (or, just as important, non-actions) in something like real time.

Prime ministerial evidence sessions with the Liaison Committee have been badly disrupted by the political upheavals of the last two years. Former Prime Minister David Cameron appeared before the Committee three times in each of the 2012-13, 2013-14 and 2014-15 sessions, the last of these being in February 2015. There was then nearly a year, to January 2016, before the first of only two prime ministerial appearances in the post-election 2015-16 session, the second being achieved only with difficulty in May 2016. With Mr Cameron then gone in June, new Prime Minister Theresa May appeared only once during the 2016-17 session, in December 2016. The session scheduled for 24 April 2017 was cancelled because of the early general election. With the post-election Liaison Committee now finally up and running, at best nearly a year will again have gone by before the Prime Minister’s next appearance.

The Liaison Committee could make an important Brexit contribution simply by delivering from now on evidence sessions with the Prime Minister on Brexit that are regular and rigorous.

The Committee’s questioning should probe when, through what processes and on what bases the Prime Minister is taking critical Brexit decisions.

Given the Liaison Committee’s welcome recent shift to limiting the number of Members participating in each prime ministerial evidence session, the Committee should ensure that its Brexit probing reflects the questions that all select committee chairs want put to the Prime Minister.

But there are four further ways in which the Liaison Committee could play a significant Brexit role.

These lie within the Liaison Committee’s established remit of promoting select committee effectiveness. The Liaison Committee should aim to help select committees collectively to have delivered effective scrutiny by the end of the Brexit process.

This might involve, for one thing, ensuring that steps already underway to increase select committees’ Brexit-related staff capacity and improve information-sharing deliver the support that committees need.

The Liaison Committee might also look to minimise duplication between select committee Brexit inquiries. As of mid-November 2017, there are 25 live Brexit-related select committee inquiries in the House of Commons, being conducted by 18 committees. These include two on devolution, and three on the EU (Withdrawal) Bill, for example.

It might be argued that if multiple committees look at the same issue and reach the same conclusion, this would be more powerful than a single committee report. Equally, multiple committees reaching different conclusions on an issue could be evidence that the relevant policy arguments are genuinely finely balanced. And multiple inquiries reduce the risk of conclusions being reached on too narrow a range of evidence.

But, against this view, the existence of multiple committee inquiries into the same issue risks wasting resources, trying the patience of specialist witnesses and organisations, and appearing disorganised and confusing to the public.

A step on from minimising duplication would be for the Liaison Committee to engage in some light-touch strategic planning of select committees’ Brexit work. In the interests of collective effectiveness, the Committee could aim to ensure that at least - but ideally only - one committee produces a report on each of the major Brexit issues, timed to maximise impact.

The Commons Liaison Committee might usefully look at the practice of its Lords counterpart in this respect. In November 2016, the Lords Liaison Committee established a Brexit Liaison Group which has now met eight times, publishing a note of its activity each time. Nine Lords select committees and subcommittees are currently conducting 15 inquiries with no obvious overlap. As well as coordinating committee inquiries, the Brexit Liaison Group has held informal discussions with a range of external official and independent representatives (including the Hansard Society), to save committees time and help provide them with a common information base.

It is, admittedly, easier for the Lords Liaison Committee than its Commons counterpart to achieve inter-committee coordination. The Lords Liaison Committee only has 11 members, against probably 35 in the Commons (36 committee chairs are now officially members, but the Standards and Privileges committees might well again have a common chair). Lords select committee chairs are less likely than their Commons counterparts to be seeking to use a distinctive select committee agenda to build profile or achieve ministerial office. And a large share of Lords select committee scrutiny is conducted through the EU Committee, which is already an over-arching body with subject-specific sub-committees, and which took the lead in planning a coordinated set of inquiries shortly after the EU referendum, before the Liaison Committee group was established.

But joint working between some Commons select committees is increasingly normal. And the Liaison Committee has sometimes acted previously to reduce overlap and request joint working. There is no evidence in the Liaison Committee's 2016-17 minutes that, on Brexit, the Committee in the last session considered taking steps in this direction. But the nature of the Brexit process would seem to demand that the Committee steps up its role in this area.

In helping select committees to achieve impact, the Lords Liaison Committee has the further advantage over its Commons counterpart of the Lords’ greater ability to get select committee reports debated in the chamber, and in a timely fashion, because of the Lords’ greater control over their own timetable.

But in the case of Brexit, select committees (and parliamentarians generally) in both the Lords and the Commons face the common challenge of uncertainty - over the timing and sequencing of some key steps in the process, and over the information that the government will publish or otherwise make available, including through ministerial select committee appearances. While some of this uncertainty is probably unavoidable given the unprecedented nature of the Brexit process, some of it arises from the government’s apparently ad hoc approach, and it hampers effective planning.

This creates a further need that, in the Commons, the Liaison Committee might usefully (if belatedly) fill - namely, to act as select committees’ representative in dealing with the government, primarily in securing from it a more timely, consistent and transparent approach to the provision of Brexit information and documents.

Again, this would fit within the Liaison Committee’s established remit: the Committee already deals with the government on behalf of select committees on matters such as civil servants giving evidence, or the holding of pre-appointment hearings.

On this potential Liaison Committee role, one interesting question concerns the relationship between the Liaison Committee and the Exiting the EU Committee (the ‘Brexit Committee’). Is the latter ‘just’ a departmental select committee, or does it have an overarching Brexit role? (The question mirrors that which surrounds the role of the Department for Exiting the EU.)

A side-effect of the successful 1 November Opposition Day motion for a return, which requires the government to provide a set of Brexit impact assessments to the Brexit Committee, is that it arguably made the Brexit Committee, not the Liaison Committee, primus inter pares on the receipt of Brexit information on behalf of the Commons.

Given the timing of the motion, it is impossible to know whether the Official Opposition nominated the Brexit Committee to receive the documents because that committee has a chair from the Official Opposition party, because the MP pushing most prominently for release of the documents (Seema Malhotra) is a member of that committee (and also from the Official Opposition), because the Brexit Committee is regarded as having an overarching Brexit role on behalf of the Commons, or because the Liaison Committee did not exist at the time. Labour’s Shadow Brexit Secretary Keir Starmer told the Commons that the Brexit Committee “seemed the obvious committee”.

In this context, it is perhaps telling that it was the chair of the Brexit Committee, Hilary Benn, who met with the Lords Liaison Committee’s Brexit group at its third meeting in January to discuss cross-House coordination. And that Keir Starmer’s comment in the debate on the motion for a return came in response to a concern raised by Sarah Wollaston that all select committees should have access to the documents.

Whoever wins today’s election to chair the Liaison Committee, then, it seems the Committee has a potentially critical Brexit role – but the new chair should clarify early with his or her Brexit Committee counterpart how far this extends beyond effective questioning of the Prime Minister into an overarching Brexit remit.