The 13 pin is exactly the same wiring as the 7 pin.my caravan isn’t lighting up tail lights, but all other lights work fine.
However… while testing, I got a strange sequence of events.
I’ll also point out that the 13 pin socket has the left and right tail lights on different pins, so whatever it is, is affecting both sides.
Left and right tail lights are separate, one side also supplies the number plate light.
The 13 pin sockets have extra pins for reversing lights + supply etc.