Had a similar issue on our 2000 V70 SE. The first sign of trouble was the license plate lights flickering when opening/closing the tailgate.

The tailgate (Volvo calls it a liftgate) wiring harness is routed along the interior upper left body and feeds all the of tailgate electrical functions: rear window defroster, power latch, rear wiper, license plate lights, upper brake light, etc.

There's a black plastic cable tie that secures the tailgate wiring harness to the upper left interior body roof area before it's routed into the tailgate through the left tailgate hinge. That cable tie was fitted too tightly during car assembly, it strains the wire harness with use over time, and the wire(s) chafe and break. No melted mess on ours, but that's possible if it goes unaddressed for too long.

Solution was to butt-splice and solder all of the chafed or broken wires (there are many wires in that harness), tape and rewrap the wire harness, and get rid of the cable tie to reduce wire strain. The rear left side interior trim panels need to be removed to access and splice the wiring harness and remove the cable tie.

Going on five years now with no repeat tailgate electrical issues.