Now that I do see, but it happens with my tuner out of the game too. I assume that is the slight change in string pressure and position (on the nut), causing it to stop vibrating. It seems to happen on all of my bass strings.
I mean the note sustains fine, until I start adjusting tuning, so the string is "moving", when the sustain is cut short.
Albeit, less so, with the new Tusq nut. I bet a GraphTec would almost eliminate it.