Which of the Conservative and Labour parties is most likely to split over Brexit? Or perhaps it is more apposite to say which party will break up first, since the gravitational force of competing visions of the UK's future relationship with the EU are threatening to fracture each of them.

On my show last night the divisions in the Tory Party were on full display – with the chief secretary Liz Truss implying that the prime minister is wasting her time wooing party leaders to find a Brexit compromise, and should concentrate instead on reaching out to the 118 Tory MPs and the DUP’s 10 who voted against her.

What Truss appears to believe is that if the EU can be persuaded to either remove the backstop or put a time limit on it, the PM’s deal would pass through the Commons.

Which, for what it’s worth, is not what Theresa May thinks, according to those close to her: she has been persuaded, I understand, both that the EU won’t move enough on the backstop, and that even if it did she would not win a majority.

So she has to explore whether if she softened or gave up some of her Brexit red lines, such as that a post-Brexit UK would not have the power to negotiate independent trade deals by being in the customs union forever, a cross-party alliance would carry the day for her.

But both Liz Truss and the senior Tory Brexiter MP John Whittingdale made clear on the Peston show they would implacably oppose such a compromise.

It would only work if May is content to see cabinet resignations and even MPs resigning from the party. That is why most Tories think the idea of a customs-union fudge has been allowed to become a talking point only to distract from the absence of a strategy that might actually work.

As for Labour, it is either liberated or imprisoned by the Brexit process enshrined in that notorious motion passed at its last conference.

What was striking, again on the show, was the relative degree of unity shown byhe the shadow chancellor John McDonnell, Lisa Nandy – whose instincts are to deliver some kind of Brexit, and who has adopted Gordon Brown’s idea of citizens’ juries delivering a unifying plan – and Ben Bradshaw, who wants a referendum.

This is the calm before the earthquake. But the moment of truth, which is Labour having to choose whether or not to back a referendum, cannot be many days away.

A so-called People’s Vote is seen by the likes of Bradshaw as the inevitable and obvious destination, but by others such as Corbyn’s all-powerful aides Seumas Milne and Karie Murphy as a toxic chimera.

McDonnell told me that on Monday Labour will amend the motion laid by the PM that is supposed to outline her plan to secure Brexit – and it will call for Labour’s vision of Brexit, namely customs union and semi-membership of the single market forever.

This will presumably be the moment when the conceit that Theresa May has repositioned herself as the leader of a cross-party national government is exploded, as will the hopes of the Gaukes, Hammonds and Rudds in the cabinet who think customs-union membership is the silver bullet. Because May will presumably instruct her MPs to vote against Labour’s motion, and it will fail.

What then?

It will be the fight to the death in the shadow cabinet, between the shadow Brexit secretary Keir Starmer, who wants Labour to back a referendum, and the party’s chairman Ian Lavery, who backs Brexit – with Corbyn himself an instinctive supporter of Lavery’s position.

In terms of background pressures, much of the Momentum Corbyn-supporting movement wants a referendum and trade unions are divided – although the most powerful trade union leader Len McCluskey remains implacably opposed to a People’s Vote, I am told.

Where will it end? One senior Labour source tells me: “If Keir keeps pushing, he’ll win. Lavery is the one who has threatened to resign. He won’t though”.

There is symmetry here, because another influential Labour figure tells me Starmer will quit if he loses.

McDonnell himself may be the decisive influence.

I pressed him last night on whether those who say he is ideologically opposed to a referendum were correct – and he was very clear he is not.

He chose his words close carefully.

But those close to him say his overriding priority is to move from opposition into government, and that in the end he will take the position that yields the greatest electoral advantage, or wreaks the least harm, to his party.

With a clear majority of Labour members and supporters backing a referendum and an even greater majority opposing a hard Brexit, that leads most of those close to McDonnell to conclude he’ll ultimately side with Starmer.

He will also be acutely aware that if he doesn’t, and Labour is seen to have failed to act decisively to prevent an abrupt and chaotic rupture with the EU, the long-mooted day when centre-ground Labour MPs pack their bags and quit Corbyn’s Labour Party will be hastened.

Robert Peston is ITV’s Political Editor. This article originally appeared on his Facebook page.