New paper explains why the Universe ended up with three dimensions

I would have been interested to learn the reason, but here is a case where an article really tells no more than its headline: there exists this paper, see, that explains why the universe has three dimensions.

It's a shame, because as I was reading the article, I was impressed by how well it was putting technical matters into plain English. It's just that I was also anticipating getting to the juicy details, which it never did. Oh, okay, I thought, I guess I'll go try to read the scientific paper itself.

When I clicked to the paper, though, all I got was an abstract, even shorter than the article that I came from. To read the full article, I must subscribe to the journal. I have nothing against that --- unless the research was publicly funded.

Anyway, I just thought it was funny that the article was nothing but an exquisitely written segue, with no real content beyond its headline. On the other hand, had I got to the real content, I likely would not have understood it, because the reason that the universe is three dimensions is probably: math.

The whole thing reminds me of Shakespeare's King Lear...

Fool: The reason there are only seven planets is a pretty reason.

Lear: Because there are not eight.

Fool: Indeed, sire, thou would'st make a good Fool.

> Any number of dimensions could have worked equally well at this point; there wasn’t really any way to tell the difference between a universe with one dimension and a universe with seven.

What is that supposed to mean? Presumably you could still, if you were a being able to withstand the crazy physical conditions in the early universe, tell whether the universe had three or more or less dimensions? What's hindering such experiments in the early universe?