The immediate, if not necessarily underlying, reason why a majority of Jeremy Corbyn’s fellow Labour parliamentarians have lost confidence in his leadership is his performance during the EU referendum campaign. They reckon his allegedly lukewarm support for remaining in the EU made a significant contribution to the remain side’s defeat.

During the campaign the Labour leader put his level of support for the EU at no more than “seven to seven and a half out of 10”. Meanwhile, when the ballot boxes were opened, remain’s worst performances were often to be found in working-class Labour heartlands in the north of England and the Midlands. Unsurprisingly, Labour MPs put two and two together and pointed the finger of blame at their leader.

Yet, in truth, there is little evidence that Mr Corbyn’s campaigning efforts – or those of any other Labour politician – made much difference either way to the willingness of Labour supporters to vote for remain. A split in Labour’s ranks was in evidence from the moment the referendum was called – and the picture simply did not change during the subsequent months of campaigning.

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This can be seen if we examine how Labour support for remain evolved in four polls that not only polled immediately before the day of the vote, but also both in mid-May, shortly before the onset of the official pre-election period on 27 May, and in mid-February, just as David Cameron was concluding his renegotiation of Britain’s terms of membership and firing the referendum starting gun.

Back in February these polls – from Ipsos Mori, YouGov, ComRes and Survation – reckoned on average that just under three-quarters (74%) of those who voted Labour in last year’s general election intended to vote for remain. In short, it was apparent from the very beginning that a substantial minority of Labour supporters were disinclined to follow their party’s recommendation to back remain.

The early months of campaigning simply saw this pattern maintained. When these four polls were conducted in mid-May, again on average 74% of Labour supporters indicated that they would vote remain.

When these polls produced their final estimates on the eve of polling day, support for remain among Labour supporters had eased back a little, but still stood at 72%. Moreover, this was against the backdrop of a marked drop in the overall level support for remain, a drop that was especially in evidence in polls conducted by phone, which account for three of the four polls examined here.

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Far from being especially marked amongst Labour supporters, it was amongst Conservative voters that the fall in support for remain during the campaign appears to have been heaviest.

Earlier in the campaign our four polls suggested Conservative supporters would divide roughly 50:50 between remain and leave. In February 48% of them said on average that they would back remain, while in May the figure was 51%. However, by the time polling day was approaching support for remain had fallen back to 44%.

Of course, if Labour had fought a more enthusiastic and effective campaign in favour of staying in the EU then maybe support for leaving among its supporters might have been whittled down. But it is also open to doubt whether many of the working-class “left behind” voters that formed the core of leave support would have responded to such efforts.

In truth, if the finger of blame for remain’s defeat is to be pointed anywhere it is better directed at the prime minister rather than Corbyn. David Cameron failed to bring his party with him at all, and in the event that simply proved too much of a handicap for the pro-EU camp to overcome.