Did a search, no relevant answers. . .

i've often found the bulb under the foot or a piece of electrical tape often solves most irritating electrical issues.

There is another alternative... When my sending unit starting leaking gas, I went down to the local bike wrecker, where i was directed to a box of sending units of all sizes and types. Matched one up, installed it good as new.. $20

Problem solved. . .after consultation with FSM. . .from the answers posted from the panel of foolishness, I was glad I had the manual.

What was the fix? From what I know of the wiring the sending unit connects the circuit to ground when the tank is low/empty. I would guess a short in the wire as you said it does it at all times.