My ear drum, and the hair cells behind it, do exactly one thing: respond to changes in air pressure. If the ANC isn't reducing air pressure forces, then why do I hear so little? The only way to hear less is to have less changing forces to disturbed them, is it not?
I assume the good ones designed with lowpass and aliasing filters to block out anything high frequency, but inaudible, that might damage hearing. These are absolutely not air tight, so there's nothing static, and the lack of bone-conduction sounds from walking makes me think it's keeping lower frequency pressure very well under control.
When I asked an audiologist if there has been research about ANC causing hearing damage, he said that it would be hard to get a grant structure that would work with it, since nobody is making money from it, and there hasn't been.
In tinitus, the frequency that people hear is around 4000hz, because that is where the first major bend in the cochlea is. It means that the loud sound that you hear doesn't necessarily have to be a specific frequency to damage hearing in that frequency either.
But, considering the wavelength of 20kHz is 1cm and the ear canal is about 0.7, I can't imagine there's much room for anything but a nice planar wave heading down the canal, let alone within the cochlea.