I asked this morning which way The Independent Group would vote in a confidence motion in Theresa May’s government. Now one of its founders, Gavin Shuker, has provided the answer: to keep the prime minister in office, provided she offered a second referendum.

Shuker tells HuffPost that members of the group first made the offer to May’s de facto deputy, David Lidington, before their resignations:

“I went into the Cabinet Office about a month ago when Theresa May was in full listening mode and sat down with David Lidington when I was a Labour MP and I said, look you can have my support to pass any deal you like on Brexit – it can be Norway, it can be Theresa May’s deal, it can be Canada, whatever you want you can have my vote “But at the end of it I want a confirmatory referendum. “And do you know what? If you do that British politics is going to be shaken quite a bit if you get that through – I’m willing to extend confidence and supply to your government so it doesn’t fall over through the period of the referendum. If it’s a year, fine. “They obviously didn’t choose to go down that route. “But if you are asking me about future hypotheticals about when the maths of parliament are so febrile, it comes down to – to use the cliche – doing the right thing by the country.”

Though Lidington rejected the offer, Shuker said that it still stood in principle. He urged the government to support plans from the Labour backbenchers Phil Wilson and Peter Kyle to amend the next meaningful vote to approve the withdrawal agreement on the condition that it is put to the public in a confirmatory referendum.

The first thing to note is that it is an offer that May was, and is, bound to refuse for reasons of principle – her own opposition to a second referendum is well-known – and internal party management. To accept the Wilson-Kyle amendment would be to split a Conservative coalition that is every bit as restive as Labour. In that respect it is a non-starter.

Then there is the opposition. Some Labour frontbenchers believe that, should it come at the eleventh hour, the leadership could make its peace with voting for the withdrawal agreement on the proviso it was put to a referendum.

Optimists believe that those in the shadow cabinet who are opposed to a second vote are in fact a noisy minority and that such a plan could be spun internally as all things to all people: a vote for Brexit for those in leave constituencies, and a vote for a second referendum for those whose electorates backed remain. Failing to whip in favour of the plan would also risk a string of shadow ministerial resignations and further defections (the reverse is true for May).

But we know that a good chunk of the PLP is sufficiently squeamish about being seen to thwart or frustrate Brexit to oppose any parliamentary move to extend Article 50, especially given the widespread assumption that Theresa May will eventually have to do it herself. It is also clear that the leadership’s preference is to facilitate the passage of a deal without precipitating a further electoral event. When faced with a choice between jettisoning its frontbench Remainers and those in Leave constituencies, it has always chosen the former. Support from either Downing Street or the Labour leadership is more or less essential if Wilson-Kyle is to pass, but for these reasons is unlikely to be forthcoming.

That the Independent Group have drawn a straight line between the passage of that amendment and May surviving in office makes support from Labour unlikelier still. As far as Team Corbyn is concerned, the offer is a political gift. “Offering to prop up a Tory government that has inflicted so much suffering on our communities is shameful,” a source says. “This grubby offer shows what the new establishment coalition really stands for – austerity, privatisation and the interests of the super-rich”.

Given that the price Shuker has set for supporting a Brexit deal is too high for either leadership, it suggests the new grouping has decreased rather than increased the chances of a negotiated settlement passing the Commons – and with it increased the chances of the general election that is much too soon for it to fight.