Since parliamentary sovereignty implies that the outcome of the BREXIT referendum is not legally binding it implies that it has a “cheap talk” quality to it: even if you prefer stay you can (and perhaps you should) still vote leave. Why? The parliament can still choose to stay and the dissonance between popular vote and parliament can be used as a weapon to renegotiate terms. This explains why even reasonable intellectuals like Ambrose Evans-Pritchard side with BREXIT. If you have any grievances whatsoever with the remain politicians or the EU itself, even if you really believe in staying, you still have a good reason to vote leave: stick it to them.
You can be the Labour party and want to damage the Conservative Party, you can be a disgruntled layperson who has been dealt a bad hand, you can be for Europe but dislike the arrogance emitted from Brussels, you can dislike the elites telling you to stay, the option of cheap leave-talk to stick it to them and get better stay terms is a fully rational option.
I think the very framing of the referendum favors the leave vote just like the yes/no Greek referendum favored the no vote (which the Greek government interpreted as yes of course, qed). This might explain why the Sun sides with leave, the polls are turning as they do for leave and why Google searches show the leave side ahead.
BeLEAVING might simply be a cheap but powerful talk hence a rational option and the more likely outcome…