Tory party politics, says a leftwing character in James Graham’s new West End play, Labour of Love, consist largely of “posh squirrels fighting in a bag”. This week, the Conservatives look all set to provide spectacular proof of Graham’s character’s scathing comment. Theresa May has come to the party’s conference in Manchester with her authority broken. In June she squandered her majority in an election she didn’t need to call, leading a campaign that exposed her failings and left Brexit – and everything else on her agenda – looking even harder than before. Not since 2005 has a Tory conference been more obviously a leadership audition than it is this year. The posh squirrels can hardly contain themselves.

For the last four months, Mrs May has only been kept in office by her party’s inability to unite around a successor and by its fear of an early general election. Those concerns remain the prime minister’s biggest assets this week. Yet they have no long-term value. She occasionally hints that she is still in this for the long haul. Almost no one in her party agrees.

Party conference is therefore the biggest test since the election of any authority she may still possess. Not for nothing have her friends said that these four days in Manchester are about one thing only: survival. Judging by her tense and defensive interview with Andrew Marr on BBC1 and by Boris Johnson’s uncontrollable ego, those survival chances remain on a knife edge. Yet the haunting question remains: who would be better?

The weakness of Mrs May’s position is highlighted by the two things that she will be trying to do in Manchester this week. The first of these is to change the subject from Brexit to social policy. On Sunday she tried again to resurrect her social agenda, with announcements on student finance and on help to buy for first-time homeowners.

These announcements will surely be followed by others. Yet they fall short. They recognise the problems about student costs and housing shortage, without providing credible solutions. The suspicion remains that Mrs May made them because of low Tory ratings among young voters, not because of the deeper policy problems. They sound like throat-clearing when set alongside the cornucopia of promises that Labour, whether credibly or not, now offers to these voters on such issues. The reality for Mrs May is that she failed to flesh out her social reform agenda when she had some credibility in 2016. Trying to do it now feels simply too late and too unbelievable.

The second problem remains Brexit. The prime minister’s Florence speech tacked as emolliently as it could towards a slightly more sensible approach to Brexit. But it was lamentably cautious all the same, was unworthy of the gravity of the issues, and it managed to annoy both wings of her party. Soft Brexiters thought it too hard. Hard Brexiters thought it too soft. The attempt to face both ways remains catnip to Mr Johnson’s undisciplinable ambition. The risk is very great for the Tories that this week in Manchester will be reported through the prism of the foreign secretary’s ego and those of other pretenders like David Davis and Jacob Rees-Mogg. Mrs May made it abjectly clear on Sunday that she is both temperamentally and politically too weak to sack someone who is disloyal to her and an embarrassment to the country.

It is possible that Mrs May will get her wish this week and survive. The Conservative party has deep instincts of self-preservation and discipline. The idea that a new leader, Mr Johnson least of all, would unite either the party or the country on Brexit or anything else lacks credibility. Though they are divided on many things, the Tories are as one in their wish to avoid handing power to Jeremy Corbyn. But the real question is not whether Mrs May will survive but whether she deserves to. The reality is that there is almost no sign whatever of that.