Russia has a Losharik nuclear submarine that was tested up to 2500m (8200ft) if Wikipedia is to be believed. It got in the news when it had a fire that killed half a dozen decorated officers.
USA probably has something similar.
No doubt, depth capability is one of the big features you go after when building a combat submarine. The deeper you go, the faster you can go while retaining stealth (cavitation is the limiting factor which decreases with the added hydrostatic pressure), and, if you can go significantly deeper than your adversaries, you're more or less invulnerable to attack because they can't shoot at you.
That's also why military specs on things like this are always lies, you don't want anyone knowing how deep you can actually go, which likely includes most of the crew.
I think I read in one of the earlier Titan threads this week that density increases pretty slowly with water depth. Water isn't very compressible, after all. So, to your question, I would guess "yes, but not by much".
