Nicola Sturgeon disagrees profoundly with Theresa May about Brexit. Scotland’s first minister wants Scotland to remain in the EU single market and believes it should continue to apply the EU’s rules on freedom of movement. Indeed, she is so concerned about the prime minister’s opposition to these ideas that she wants to hold a referendum on Scottish independence so that the country can determine its relationship with the EU for itself.

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Behind this disagreement lies a presumption – that whatever voters in England and Wales might want out of Brexit, people in Scotland want something very different. After all, did not voters north of the border vote by 62% to 38% in favour of remaining in the EU, whereas in England and Wales there was a 53% to 47% vote in favour of leaving? That would seem quite clear evidence of a very different attitude towards the EU north of the border.

Yet to date little effort has been made to check out this assumption by asking voters in Scotland what kind of Brexit they would like to see. New research published today by NatCen Social Research finally does so – and makes rather sober reading for Scotland’s first minister.

It turns out that Scots are not so keen on freedom of movement after all. As many as 64% believe that, post-Brexit, anyone from the EU who wishes to live in Britain should have to apply to do so in the same way as anyone from outside the EU. Even more, 72%, think that the same rule should apply to any British citizen who wants to go and live in the EU.

Unsurprisingly this mood is most prevalent among the minority of Scots who voted to leave the EU, more than 80% of whom would like to put migration between the EU and Britain on the same footing as that between anywhere else in the world and Britain. But it is also relatively widespread among those who voted to remain, more than half of whom take the same view.

Most voters in Scotland might have voted to remain, but that does not mean that they are so enamoured of the merits of the EU that they necessarily wish to maintain freedom of movement now that the UK is heading for the EU exit.

That is not to say that Sturgeon has misread voters’ mood entirely. No fewer than 93% express support for allowing EU companies to trade freely in Britain and for ensuring that British companies are able to do trade equally freely across the EU. Even leave voters think this would be a perfectly sensible outcome. But of course what this does mean is that what voters in Scotland want out of Brexit is closer to what the prime minister has in mind than Sturgeon’s vision of what should happen – that is, ending freedom of movement but securing an “ambitious and comprehensive free trade agreement”. It also means, by the way, that attitudes towards Brexit in Scotland are very similar to those in the rest of the UK.

It is, thus, perhaps not surprising that there is relatively little support for the Scottish government’s idea that Scotland might have a closer relationship with the EU even while still being part of the UK. As many as 62% say the rules on immigration into Scotland from the EU should be the same as those for immigration into England and Wales. Only 25% back the idea that it might be easier for someone from the EU to migrate to Scotland than to England or Wales. Equally, 62% say the rules on trade between Scotland and the EU should be the same as those in the rest of the UK, while just 34% think there should be a more liberal regime north of the border.

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Between them, these findings raise severe doubts about the wisdom of the Scottish government’s decision to turn a disagreement about what Brexit should mean into the crux of an argument as to why Scotland should have a second opportunity to back leaving the UK. The level of commitment to the EU in Scotland may be broad but it is also seemingly too shallow for Brexit to be an issue that is likely to change many minds about the merits of independence. Even among those who were already in favour of leaving the UK in the first independence referendum, just over three in five would like to see an end to freedom of movement.

Perhaps what Nicola Sturgeon is anticipating is that the combination of free trade and immigration control that most Scots want and Theresa May hopes to achieve will prove impossible to secure, and that then voters north of the border will want to seek independence. But at the moment that looks like a very big call for her to make.