Are you saying that this A-theory predicts that there is a preferred foliation?

Not to speak for pragmatist, but, yes, that is my understanding. But, importantly, the foliation isn't just preferred by some distinguishing physical characteristic (the way a preferred reference frame would be, for example). Rather, the foliation is preferred in a more ontologically fundamental sense: When one leaf exists, no other leaves of the foliation exist at all, nor do the parts of spacetime that they would "foliate". For the presentist/A-theorist, at this moment, a completely exhaustive ontology of the world contains nothing that is not in the present leaf.

By that logic, wouldn't B-theory predict that no foliation is possible at all? Or that all foliations are equal, whether they are timelike, null or spacelike? If so, the B-theory has been clearly falsified (if you can ever falsify anything in philosophy of physics).

The B-theory allows foliations to be different from one another in physically real ways. The B-theory doesn't allow that leaves of one special foliation "pass into and out of existence", which is what the presentist/A-theoretic approach requires.

(That is my understanding of what a presentist would say, anyway. But, as I said, I can't really make sense of presentism, so I might not be portraying the view accurately.)