This is not what he was suggesting. The jack needs to be tested completely isolated from all other parts.
You need to remove the white wire from the center pin, then measure the resistance between hot and ground of the jack. Place one probe on the center pin and the other on the ground tab. Resistance should be over range, or the continuity should not beep. If this test fails, the problem is the jack, and the problem either needs to be found (debris inside the top of the jack, or something bridging on the bottom side that was not visible on the images shown) and fixed or the jack replaced.
The problem certainly could be the wiring at the pot; we have not seen any pictures of it so it is hard to say. It would be very difficult for the problem to be the braid.