1. A contest without Burnham

Greater Manchester mayor Andy Burnham, the likely favourite among Labour MPs, is not in parliament. That means any resolution in the coming weeks would require a sitting MP to trigger a contest by getting the backing of 80-plus colleagues, or persuading Starmer to set a timetable to quit.

The MP for Makerfield in Greater Manchester, Josh Simons, announced on Thursday afternoon that he is giving up his seat and wants Burnham to take it. But it remains to be seen if Downing Street will let him stand, and if he did, whether he would win.

If that did not happen for any reason, then there could be an impasse. Wes Streeting, who resigned as health secretary on Thursday, had reportedly pledged to start a contest but seemingly failed to raise the necessary support among MPs. Other possible challengers – like Angela Rayner and Ed Miliband – do not seem to be mobilising fully.

But things could change quickly, for example if one of the promised new tranches of documents about Peter Mandelson’s appointment as ambassador to the US brings more damaging revelations.

It is worth noting that, if someone did launch a formal challenge, Starmer could fight them – and he could then win a vote of Labour members.

2. A contest with Burnham back

If Downing Street allows the Labour machine to endorse Burnham as the candidate in Makerfield, it would take some weeks for the byelection to happen. A win would send him back to the Commons.

Letting Burnham run as an MP is possible. While Downing Street made sure he was blocked from standing in the Gorton and Denton byelection, Starmer would be under huge pressure to change course this time around.

Again, if a leadership contest followed, Starmer might opt to fight and could win. The party uses the single transferable vote system, where members rank candidates by preference. It would not be inconceivable for Starmer to win a three-way race based on second-choice votes from people who did not want one of the alternatives taking over.

3. An even slower Burnham contest

This could be different purely from a timetabling point of view. If Burnham is rebuffed in his attempts to be selected for Makerfield, something could emerge in the long term.

There is also a world in which Downing Street agrees to allow Burnham to fight a Commons seat once his mayoral term ends in 2028. Some MPs believe changing prime ministers nearer an election might be better, to provide a sense of freshness and momentum.

The challenge with this idea is that it would leave a deeply hobbled lame duck PM in Downing Street for two years, struggling to implement an agenda and very possibly driving Labour even lower in the polls.

4. Starmer survives

Hang on, what?

Yes, it does seem unlikely: Starmer has proved an underwhelming prime minister by most metrics, and while close to 100 Labour MPs have called for him to go, quite a few more are known to privately want this to happen.

But at the same time, someone has to be prime minister, and if Starmer does not quit – which, thus far, is not on the cards – then he will need to be pushed. Burnham might not win a byelection; none of Streeting, Miliband or Rayner might go for it.

This would feel like a recipe for managed decline. But there is, in theory, the possibility of Starmer reinventing himself and sending Labour back up the opinion polls. That maybe feels like the least likely outcome of all.