Yesterday, the Guardian HE Network ran an article of mine under the headline ‘University vice-chancellors, start calling for a second EU referendum’.

Many interpreted this as a call for a straight re-run of the 2016 referendum on the UK’s membership of the EU. They were wrong to do so, as this version (which is exactly as it was submitted to the Guardian) shows. I make no complaints about the editing that was made, even though half my words disappeared.

Yet, hopefully, the argument will be clearer in this full version. A second referendum would take place in different circumstances at a different time with different options – though could well have a similar result.

The higher education sector controversially followed one strategy in the 2016 referendum on the UK’s membership of the EU and a completely different strategy in the 2014 referendum on independence for Scotland. On Scotland, vice-chancellors (with the odd rare exception) stayed out of the fray; on the EU, they jumped in with both feet.

The most obvious manifestation was the pro-Remain Universities for Europe organisation. Some, like the journalist Michael Crick at the group’s launch, complained about university leaders getting their hands dirty – he suggested the event was ‘bogus’, ‘unhealthy’ and ‘un-academic’.

But it is the core function of all vice-chancellors to fight for what they perceive to be in the best interests of their institutions. Moreover, it is possible to believe students should be exposed to a range of views and that vice-chancellors were right to express their (one-sided) concerns publicly.

One reason why this is poorly understood is that the vice-chancellors’ club has a name – Universities UK – that sounds like it is a mouthpiece for all staff and students. For the first 82 years of its 99-year life, the organisation was called the Committee of Vice-Chancellors and Principals. This conveyed its role better. (I am not suggesting it should mark its 100th anniversary by reverting to the old name. That ship has sailed – today, there are Universities Australia, Universities Canada and Universities New Zealand too.)

So the most taxing question about university leaders and past referendums is not why they spoke out on Brexit; it is why they said so little on Scottish independence. Yet, given the EU question will remain the number one political issue for the next four years, the question today is: what should vice chancellors do now in terms of lobbying on existential European matters?

Since the EU referendum, university leaders have worked hard to accept the result while highlighting the challenges ahead on the issues that matter most to the sector. In response to concerns that the vote for Brexit showed a disconnect between universities and their local communities, some have also responded by seeking to engage with those outside their institutions more deeply. But they have, seemingly, shied away from becoming too involved in the wider public debates. So it is worth asking whether the time has come to re-enter the fray.

In particular, it seems to me that the higher education sector’s leaders could be helping to prepare the ground for a second referendum, on our future relationship with the EU. There is a long tradition of countries having two votes on controversial European matters and it seems increasingly plausible that we could have our own second referendum on EU membership for three reasons.

First, if the Brexit negotiations go positively and we end up outside the EU but with a closer relationship to the EU than any other independent country, some will say the referendum result had not been respected. Conversely, if the negotiations go badly and we end up with ‘no deal’, this too could look different to what many people expected when they voted on 23rd June 2016. The end result of the negotiations could also be somewhere between those two outcomes of a good deal and no deal – perhaps a messy and uneasy compromise, satisfying few. In other words, all three possible options at the end of the Brexit process are unlikely to satisfy the entire 52% that voted Leave, let alone the 48% that voted Remain. A new split is likely to occur, between those who want to accept whatever is on offer and those who want to reject it all for a return to the status quo ante. In those circumstances, it would be odd to think the 2016 result should be the final word.

Secondly, the UK’s membership of the EU is an issue of such profound importance that it would be strange not to put the alternative, when we know more about it, to people in a vote. Aside from the referendum on EU membership, the UK has had 11 official referendums in less than 50 years. Arguably, not one of them was on an issue as important as the outcome of the Brexit negotiations. Moreover, we know the Brexit side of the argument would have called for a second referendum had they lost, but that would have been a straight re-run; a second referendum after the current negotiations would be on a different – more practical – question.

Thirdly, both major political parties (and some minor ones) are riven down the middle on Brexit. The two biggest ones are nominally in favour of it (while disagreeing on many of the details), but many of their members and supporters feel differently. Should a fervent Brexiteer vote Labour or Tory; ditto a Remainer? No one knows. So the common argument that our future relationship with Europe can be settled at a general election instead of a second referendum does not work – unless of course all Brexiteers are to vote for Ukip, a party that currently has no MPs, and all Remainers are to vote for the Lib Dems, who have just 12 MPs.

The most important issue for the university sector is that campaigning for a second referendum could conceivably increase the likelihood of a better outcome. Having a second referendum hanging over the heads of those on both sides negotiating the UK’s exit could do two things.

First, it could provide Brussels and the other EU countries with a roadmap for keeping the UK in the EU, encouraging them to offer us more positive options for the future. If only a very poor deal were to be on offer at a second referendum, that would surely increase anti-EU feeling and affect the result accordingly. Some people I have discussed this with tell me the precise opposite would happen: the promise of a second referendum, they say, would force the EU to offer us a truly awful deal, so that voters feel they have no option but to stay in the EU after all. Yet it seems unlikely to me that voters would wish to stay in a body that, in those circumstances, would look like it had just tried to harm their country’s national interests – especially when a ‘no deal’ option is on the table as an alternative too.

Secondly, offering a second referendum could give further cause for the UK negotiators to secure a good deal – if they do not, their deal (or any no-deal situation) risks losing at the ballot box, sending them to political ignominy. Moreover, the Brexit issues most bothering the higher education sector – such as staff and student mobility, EU research funding and regional development cash – could be placed centre-stage in the negotiations if universities lobbied confidently for a second referendum.

There is no guarantee at all that a second referendum would overturn the first one. But those backing a second referendum, irrespective of whether they voted Leave or Remain in 2016, could have helped to build a more propitious new environment in which to work and it is surely better to set the country’s future path on the basis of a referendum on a detailed proposal than on the last referendum, when people were not sure precisely what they were voting about.

Let us not forget that Dom Cummings – who was the Director of Vote Leave and the person who probably did more to ensure the UK voted for Brexit than anyone else – wrote exactly a year before the UK voted to leave that: ‘a NO vote does not mean we would immediately leave and it seems likely that the parties will be forced by public opinion to offer a second vote’.