I have increased the error window to the point of significant slop before the servo tries to correct, and gains down so low the servo would be unusable. Neither had any effect on the 10.6khz noise.

I will again note that movement or load have no discernable impact on the noise. While I still don't claim any expertise in servos, I don't see how dithering or oscillation noise wouldn't be reactive in some way to what the servo is doing?