It gives me no pleasure to report this of my former Daily Telegraph colleague, but some people who know Boris Johnson don’t trust him. Whatever the Prime Minister’s other virtues, he is not seen by some acquaintances as a man who will always keep his word, who always does the things he says he will do.

Polls appear to suggest that the public isn’t much more impressed with Johnson’s integrity. YouGov reckons just 24 per cent see him as “trustworthy” and the same proportion rate him as “honest”.

That should be a problem, given that so much of Johnson’s political strategy (and possibly Britain’s future) now rides on his ability to do the thing he has repeatedly said he will do and take Britain out of the European Union on 31 October – do or die, dead in a ditch, etc etc.

I won’t rehearse here the arguments about whether that is actually possible or likely, given EU reactions to the PM’s proposed Brexit deal and in light of the Benn Act’s fairly clear instructions to the executive.

Instead I’m more interested in some new polling that tests whether voters believe Johnson’s Brexit promise. It was carried out by Public First and asks this question:

Do you think we will leave the EU at the end of October, as the Prime Minister has pledged?

The headline figures are that 36 per cent believe we will leave, 39 per cent don’t think we will, and 25 per cent don’t know.

But dig into the numbers and some properly interesting things emerge. Among those who currently say they will vote Conservative at the next election, 53 per cent say they believe we will indeed leave on 31 October as the PM promises; only 22 per cent of current Tory supporters think he’ll fail. Those backing the Brexit party, meanwhile, are even more inclined to trust Johnson: 55 per cent of them think we are leaving at the end of this month.

The regional breakdown is interesting too. 47 per cent of people in the north-east of England believe Johnson will deliver on his promise, well above the national figure of 36 per cent. In Scotland, just 28 per cent trust the PM to take Britain out by 31 October.

What to make of this? First, I think those figures illustrate that among his chosen audiences, Boris Johnson has been pretty successful in landing his key message, and in persuading them that whatever they may think of his general trustworthiness, he will make good on that Brexit promise. That success is surely a key factor in the improving poll ratings of his party.

Second, confidence in Johnson’s ability to deliver Brexit is quite resilient. The Public First fieldwork was done last week, after the passage of the Benn Act, the Supreme Court judgement and the various other excitements that left a lot of people at Westminster convinced the UK will not leave on 31 October. A lot of Johnson’s voters either haven’t noticed those things, or don’t believe they will stop the PM delivering.

But this raises questions, one above all others: having put such faith in Boris Johnson to “get Brexit done” by the end of the month, how will all those Conservative supporters respond if he does not do so?

No 10 is preparing to attempt to shift any popular blame for a delay onto others. But the stakes could not be higher. Those numbers suggest that in such an event, there will be rather a lot of disappointed Tory voters to persuade to be angry at someone else. So far, the Johnson charm has been the decisive factor in this latest act of the Brexit drama, but if the UK is still a member of the EU come November, the PM’s ability to get away with breaking his word will have its toughest test yet.

On the other hand, if he can indeed break that promise to “get Brexit done” without losing a huge chunk of his supporters, who would bet against him at a subsequent general election? And which opposition party will rush to trigger an election fight with a man who could survive breaking such a promise?