Is there a correct gender-neutral, singular pronoun ("his" versus "her" versus "their") ?

This is still disputed and I think we see a language at change (yes, languages are always changing!) so the speakers can try out many options, and perhaps we will have an agreement in some years.

The consensus so far seems to be the so-called "singular they" (that's the one I personally use) : the verb is in plural, but the referent in singular, much like the "you".:

The student went to their room and they closed the curtains.

Now, one person can stands and they wave: note how here the referent is the same but in one case you use plural and in one singular: it's still correct!

so the pronoun is "they" - the singular they