It is entirely possible that the volume pot has an open along the resistive path in its LS section, wherein the circuit completes when the pot is at or near its lowest volume position, but as soon as the control is moved away from that position, continuity is lost between the wiper and the resistive element. This can be checked by tracking the resistance at the control as the pot is exercised. Depending upon the design of the pot, it MAY be possible to open it up and physically examine its working parts and maybe even correct any open found.
-ChrisP