It is the CHANGE in volume that affects the pitch
Since you make no mention of frequency (group delay). I take this to mean that you suspect that the absolute signal level changes the propagation delay in the circuit. This is a non-linearity that should cause IMD distortion and it should be measureable. (Signal level or amplitude is modulating the frequency of output - creating new frequencies. For example a cymbal high frequency sound riding on top of a large oscillating bass guitar note - as the bass guitar signal goes up it changes the of pitch of the cymbal HF sound - giving side bands)
FWIW - this is already a huge problem in speakers and especially those with overhung voice coils that are not particularly linear throughout their driver excursion.