It's true he is a dead man walking. Even if he clings on to power now he is very unlikely to make it through as Labour leader to the next election. The question is more when it is most expedient to replace him.
If a new challenge comes from a backbencher could well be instigated behind the scenes by one of the cabinet hopefuls, but no one will no for sure.
As a cabinet minister there is great risk in kicking off the challenge. Your job is almost certainly toast under that prime minister, and the prospect of better jobs/future promotion. You might also be seen as being unreliable in the party.
However if you can persuade a backbencher to kick off a leadership challenge, then you can de-risk your own challenge by then standing because "you had to once the leadership process was underway".
You might ask, what's in it for the backbencher they will almost certainly lose the challenge and the answrer is probably a plum job in a future cabinet or similar favour at some point in the future.
Cabinet members are probably weighing up whether it is a good idea to get their own challenge underway before Burnham gets on the scene, because he is a strong candidate and will limit their own chance of success.
Of course, the general public often sees all this behind the scenes manipulation as a bit distasteful. But it is all par for the course in politics as every politician knows, and it's fairly certain that even though the current incumbent will say negative things about the process of challenging, they will have course conveniently forgotten that they engaged in similar subterfuge during their own rise to power.