The cabinet’s summit at Chequers today is as big a test for Brexiteer ministers as it is for Theresa May. If they are to challenge her blueprint for a soft Brexit, Boris Johnson, David Davis and Michael Gove will need to come up with a credible alternative of their own.

One cabinet supporter of May’s plan told me: “The Eurosceptics sit round the table shooting everything down, but never put anything up. So far, they have produced no ideas of their own. The prime minister is proposing a sensible Brexit that most Tory MPs will support.”

The eurosceptics can no longer accuse an indecisive May of not having a plan. Her 120-page draft White Paper is on the table at Chequers. With May veering towards a Norway-EU style relationship by aligning UK goods and agricultural products with EU regulations, Brexiteers may have to come off their own fence and propose a Canada-EU style free trade deal, which May fears would harm British business.

Despite the council of war held by seven Brexiteer ministers in Johnson’s spacious Foreign Office lair last night, May should stick to her guns. If she allows the Brexiteers to dilute her White Paper, it will make it even harder to persuade the EU to accept it as a basis for negotiations. That is already going to be difficult enough. “A single market for goods is not an option,” one Brussels source said. “We have been telling the UK that for two years.” However, May hopes the EU will engage after the White Paper’s publication next week, even if agreement will then require concessions by both sides.

As well as squaring the cabinet and the EU, May also has to secure parliament’s approval for a withdrawal agreement. She is warning her ministers that if they do not endorse her compromise on customs today, MPs will likely vote for an even closer customs union. In a remarkable coincidence, the government has tabled votes in the House of Commons on this for 16-17 July.

May’s decision to finally come down on one side of the Tory divide has huge implications. Despite her hard Brexit rhetoric since the 2016 referendum, she is now advocating the soft Brexit that became inevitable once she lost her overall majority last year.

While much attention is on the Tory divide, tensions within the Labour Party on Brexit will become very important if an EU deal is struck. Jeremy Corbyn hopes to paper over the cracks by whipping his MPs to vote against the deal – even if they do so for different reasons. But May’s conversion to soft Brexit will make it harder for Corbyn to cajole all his MPs into one division lobby.

There are already four Labour tribes: left-wing eurosceptics like Corbyn who always saw the EU as a capitalist club, centrist Labour eurosceptics who oppose free movement, passionately pro-EU MPs who will do anything to frustrate Brexit and Remainers turned Re-Leavers, many of whom represent constituencies which voted Leave. May’s move potentially creates a fifth Labour group: Remainers who reluctantly accept Brexit must happen but are no fans of Corbyn and might therefore support May’s soft Brexit. Significantly, some have already been quietly sounded out by Tory ministers. “If she gets a soft Brexit deal, I think I would vote for it,” one Labour MP told me. Another said: “A lot of us would probably support a sensible Brexit, whatever our own party told us to do.”

Trips to Chequers: a history of famous visits Show all 14 1 /14 Trips to Chequers: a history of famous visits Trips to Chequers: a history of famous visits David Cameron hosts Chinese President Xi Jinping in 2015. A notable visit if only for the pictures of David Cameron looking awkward drinking a pint AFP/Getty Trips to Chequers: a history of famous visits Edward Heath and the Queen host then US President Richard Nixon and his wife at Chequers in 1970 PA Trips to Chequers: a history of famous visits Former Soviet heavyweight Mikhail Gorbachev visits Margaret Thatcher at Chequers in 1984 Getty Trips to Chequers: a history of famous visits David Cameron hosts German Chancellor Angela Merkel in 2015. Mrs Merkel had previously visited during Gordon Brown's time as PM AFP/Getty Trips to Chequers: a history of famous visits Gerry Adams and Martin McGuiness of Sinn Féin visit Chequers in 2003 PA Trips to Chequers: a history of famous visits President of the European Commission Jean-Claude Juncker visits Chequers in 2015 to renegotiate the UK's EU membership Getty Trips to Chequers: a history of famous visits Prince Philip visits in 2014 and plants a tree with David Cameron Getty Trips to Chequers: a history of famous visits Then US Secretary of State Hillary Clinton is hosted by Gordon Brown in 2009. Mrs Clinton had previously visited in the 1990s as First Lady of the United States Getty Trips to Chequers: a history of famous visits Russian President Vladimir Putin is received at Chequers by Tony and Cherie Blair AFP/Getty Trips to Chequers: a history of famous visits Margaret Thatcher hosts Zimbabwean President Robert Mugabe in 1988 PA Trips to Chequers: a history of famous visits George Bush Sr. visits Thatcher at Chequers in 1984 PA Trips to Chequers: a history of famous visits President Dwight D. Eisenhower visits Harold Macmillan at Chequers in 1959 PA Trips to Chequers: a history of famous visits Ramsay Macdonald hosts W.T Cosgrave, President of the Executive Council of the Irish Free State and Sir James Craig, Prime Minister of Northern Ireland , at Chequers in 1924 to negotiate the Irish border PA Trips to Chequers: a history of famous visits David Lloyd George, the first Prime Minister to use Chequers, hosts French Prime Minister Aristide Briand and Marshall Foch in 1921 Getty

This tribe would come under enormous pressure to toe the Labour line and oppose May’s deal and might well be threatened with deselection by their local parties. Corbyn believes this Commons’ vote will offer Labour’s best chance of forcing a general election before the one due in 2022. But his internal critics point to the Fixed-term Parliaments Act 2011, claiming that rejection of the deal by the Commons would force a Tory leadership election (as May would resign), but not a general election (as Tory MPs and the Democratic Unionist Party would not vote for one in the separate vote required by the act).

Labour’s soft Brexiteers could outgun a last-ditch attempt by hardline Tory eurosceptics to scupper May’s deal by joining forces with the Labour leadership to vote it down, in the hope that results in a “no deal” exit next March. So it is possible that May secures a Commons majority for her EU deal with the help of Labour MPs.