I had the same problem on the near side with the earth having burnt out, melting the multi pin connector and essentially disconnecting the earth by melting it off the plug. It looked like someone had crushed the pin in the plug at some point causing a poor connecting which then arc'd out over time and did the damage.
I ended up soldering an additional earth onto the light circuit board - they're just metal channels on the plastic back plate to the light cluster - fed that through to the car and soldered a new bullet connector onto the car side of the wiring loom, so the earth is on it's own bullet connector and properly insulated. I removed the earth from the multi plug on the cluster and just plug it in as normal to the light unit, it just now has a slightly separate but still able to disconnect earth connection.
Not ideal but they're odd pins that you can't easily obtain and having melted the connection on the circuit board a new pin would have been useless anyway.