At around midday on Tuesday, Conservative party members attending their conference in Birmingham will face a dilemma that says a lot about the terribly divided state their party is in. In the main hall, home secretary Sajid Javid, seen by many as a contender for the leadership when Theresa May departs, will be limbering up for his keynote speech.

It will, no doubt, be portrayed in the media as Javid’s pitch to the grassroots for the top job. But not far away, outside Hall 1 of the International Convention Centre, the queues will have been forming for hours for another lunchtime event, one that will perhaps have even greater significance for the future direction of the Tory party.

“I would get there early,” says Paul Goodman, a former Tory MP and now editor of ConservativeHome, which is hosting a speech by Boris Johnson due to begin at 1pm. “I think Sajid Javid’s address will be very important,” adds Goodman. “But people are saying that the two biggest events of conference will be the Boris Johnson speech, and the prime minister’s on Wednesday. And I would go along with that.”

Last week, Labour’s divisions over Europe and Brexit were on full display in Liverpool. But, worryingly for the Tories and encouragingly for Labour, Jeremy Corbyn’s party managed them pretty well and emerged looking, on the surface at least, like a force united around a radical programme for government.

Tory cabinet ministers now admit they are deeply worried because Labour seemed so hungry for office and so united in its determination to fight an election soon – and win. By contrast, they are all too aware that their own party has spent the past two years doing little but obsess about Brexit. They know they lack a counter-offer anything like as well formed as Labour’s.

In a pre-conference interview with the Observer, the health secretary, Matt Hancock, recognises today that his party must get its act together soon or the result, for them, will be catastrophic. He says it is “mission critical” that Tories, starting this week, “win the argument for a well-run market economy” and defeat Corbyn’s “scary” leftwing agenda, which “could really sink Britain”.

Boris Johnson’s speech is expected to be one of the biggest events of the conference. Photograph: Victoria Jones/PA

The challenge for Theresa May, therefore, will be more than just containing the vicious party war on Brexit, and the leadership speculation that feeds off it. She also needs her cabinet to stay together sufficiently to spell out a coherent domestic programme that can convince people her government is about more than delivering – and rowing about – Brexit.

It looks a daunting task. On Europe, most of her party members and MPs believe her Chequers plan is dead. As one senior Tory MP put it: “I have no idea how she convinces people she is the one to steer us to dry Brexit land when almost everyone rejects her route map.” As if she will need reminding of the extent of her problem, on the fringe Johnson, David Davis, Jacob Rees-Mogg and other darlings of the hard Brexit right will be rubbishing Chequers all week.

There does, however, seem to be a twofold conference plan to keep the May show on the road. First, cabinet ministers have all been ordered to come to Birmingham armed with eye-catching policy ideas to ensure delegates and the media spend some of the time, at least, on ideas, and policies. Today, the health secretary, in this paper, announces plans for the first ever national guidance on social media use by children because he is worried about the effects of excessive use on their mental health. The prime minister today reveals plans to bring in higher stamp duty for foreign buying of UK property. Housing will be a central theme of conference, whose slogan is “Opportunity for All”.

Second, there is a big Rubbish-Boris-and-stand-by-Chequers-if-we-possibly-can operation, led by No 10. Yesterday, Alan Duncan, Johnson’s former deputy at the Foreign Office, dismissed his boss of a few weeks ago as a “spent force”, “not intellectually focused”, and of “untidy mind”. Hancock and the rest of the cabinet, meanwhile, are insisting Chequers is the best Brexit plan because it is the only credible one anyone has come up with.

In reality, though, few at the top of the party have much faith that it will go smoothly. As Johnson prowls the fringe, Brexit moderates in the cabinet, appalled by the idea of him ever leading the party, privately worry that the EU’s rejection of May’s Chequers plan in Salzburg earlier this month, and the prime minister’s riposte from Downing Street, in which she ruled out possible compromises with the European side, have put her in an increasingly impossible position.

Opinion polls, including today’s by Opinium for the Observer, suggest she may have got a bounce from her “no surrender” message. But cabinet moderates fear she is boxing herself into the no-compromise line.

May’s speech on Wednesday presents her with difficult choices. Does she play the patriotic card again and please the crowd by vowing “No Surrender!”, taking on Johnson at his own game? It would be the easy way out in the short term and would appeal to the grassroots. But with a critical EU summit just a fortnight away, moderates caution that it may not be so wise to bash the Europeans too hard if she wants a good deal for the country.

As one former cabinet minister put it, the whole process is “on a hair trigger”. One false move and the Brexit talks could collapse, and with them May’s premiership. The Observer has spoken to Tory MPs on both sides of the EU debate who think she should go the other way and back a soft, Norway-style Brexit in the short term – handing a new leader the time and space to construct looser trade deals that could work for the long term.

“For most of the public, they want everyone to stop talking about Brexit – and those who voted Leave just want to know we’ve left,” said a former minister. As she headed to Birmingham yesterday, May was being pulled in all directions by her own party just at the time when unity was more important than ever.