Theresa May’s new cabinet has now rallied behind her Chequers plan, set out fully in the government’s white paper on future UK-EU relations. However, far from settling the Brexit debate, recent events have given rise to another nightmare scenario that is only just beginning to take shape: that every conceivable Brexit outcome may now not command a parliamentary majority.

The conventional wisdom in Westminster is that since the general election last year, there is no House of Commons majority for a hard Brexit. With a working majority of only 13, including the Democratic Unionists, it would take just seven Tory MPs to oppose it. But there are at least 20-30 pro-European Tories minded to do so. Yet May’s softer Brexit blueprint has also significantly increased the prospect of Eurosceptic Conservative MPs voting against her EU deal when it is put to parliament later this year.

In the febrile atmosphere at Westminster this week, there have been rumours that up to 70 Tories could oppose it – especially if, as seems likely, May makes further concessions in order to win the EU’s backing for a bespoke deal, instead of having to choose between a Canada or Norway-style agreement. Hints in the white paper about a preferential system for EU migrants, despite May’s rhetoric about ending free movement of labour, will fuel the Tory revolt.

May’s embrace for a softer Brexit has therefore changed the Commons arithmetic – and the political calculations that come with it. It is now Labour MPs, rather than Tory ones, who may prove critical. In recognition of this, May has been reaching out to Labour MPs in the hope that soft Brexit supporters vote for her deal, neutralising the impact of the Eurosceptics voting against it. In an unusual move, David Lidington, the Cabinet Office minister and May’s de facto deputy, briefed Labour (as well as Liberal Democrat and SNP) MPs on the Chequers plan.

But Labour won’t want to save May. Their leader, Jeremy Corbyn, will almost certainly whip Labour MPs to oppose May’s deal, in the hope that the ensuing chaos will result in an election. (Labour’s six tests, which include replicating the “exact same benefits” of the single market and customs union, are designed not to be met). Although some Labour MPs are no fans of Corbyn and would be tempted to vote for May’s deal if it offers a sufficiently soft Brexit, they would be branded “traitors” and potentially threatened with deselection if they helped a Tory prime minister avoid an election that Corbyn desperately wants. So, it is highly unlikely that there would be enough Labour MPs to balance out a rebellion by Tory Eurosceptics.

If neither hard nor soft Brexit can win a parliamentary majority, no-deal can’t either. Hardline Brexiters privately hope that such parliamentary gridlock would allow them to run down the article 50 clock to an accidental exit next March on World Trade Organisation terms. But MPs in all parties say they would intervene to prevent such an economically damaging no-deal outcome.

Chequers and the white paper have helped crystallise the problem the government will face in autumn: there is no majority in the Commons for the softer Brexit now advocated by May; the “Canada-plus” free trade deal favoured by the Brexiters or the no-deal exit next March favoured by hardline Eurosceptics. In these circumstances, a cross-party alliance would probably call for the two-year article 50 process to be extended beyond 29 March 2019, to which the EU would agree. But there would be no guarantee that further negotiations would produce an outcome parliament could support.

Winning the support of the cabinet and the EU was already proving difficult enough. But May’s biggest – if not insurmountable – hurdle now clearly lies at Westminster.

• Mujtaba Rahman is the managing director for Europe at Eurasia Group, a political risk research and consulting firm