I avoided to say 'at the singularity' because the predictions of the vacuum solution seem to be wrong there. But even though I think it's still wrong to talk about "compression" refering to the quantum gravity regime. Most probably nobody can tell what happens to matter when it reaches the center of a black hole. And thus classical notions like 'tidal forces', 'compression' and perhaps even 'heat' or 'spacetime' might not be applicable anymore, because we don't know the state of matter under such conditiones.No, it isn't. A black hole is a vacuum solution.
I'm reasoning sometimes whether the state of matter in the center of a black hole should't be similar to what one would expect for the state of matter in the early universe prior to inflation as it seems that matter (whatever this means in terms of quantum gravity, matter in the classical sense has existed before and during the gravitational collapse has happened and after inflation) is within a very small volume in both cases. But am not sure at all, if this makes sense.