I'm not so sure. Leaving the EU has been on people's minds for a decade + and support has grown, not shrunk. A lot of people are in incensed by Westminster and harbour a simmering anger that they aren't being listened to. That anger occasionally boils over into acts of violence.
If the Brexit vote is ignored, watered down or reduced to an insulting fudge, it will fuel people's anger and they are unlikely to forget that hat they have been betrayed in the most appalling manner possible, in the voting booth. There are many bad things on the horizon in the EU, it's not the eutopia you bicycle salesman perceive. Italy's banks are in real trouble, Portugal is virtually bankrupt, Spain has huge youth unemployment, there is almost certainly sufficient support for an EU exit in France and Holland, Greece's troubles haven't gone away and the migrant crisis, which is causing tangible anger in Germany is still with us. If the German, French and Holland population were allowed a chance to vote, they would likely quit the EU. The EU leaders have their thumbs desperately pressed into the end of the hose pipe, but pressure is building!
We are now blessed with a PM who no one has voted for. She wasn't elected at the General Election and many of her own MPs didn't even vote for her. If she ignores Brexit, every single thing that goes wrong for the UK will be blamed on her, the undemocratic nature of Westminster and our continued EU membership. Rather than go away, anger and tensions will rise and support for Brexit will intensify.
Add to all of this that the Labour Party are destroying themselves, where are all these angry people who feel cheated, let down and a seething resentment that the EU is the source of all their trouble, going to turn to? Who will represent their views? This is very rich soil for extreme politics to grow in. Your statement at the top of this post is manure in two senses of the word. Firstly it is just that, manure or dung and secondly it is further nourishment for for the soil in which extreme politics will grow and blossom.