@ Daedalus & thread
I've successfully replaced the left button, but Daedalus, you seem to be right that additional wires are required for the third button.
The cable has all 9 wires present, but for some reason the middle button pins on the PCB seem to connect to ground and nowhere - not to pin 5, which seems to the middle button pin if the pinout I'm reading is to be believed.
Strange.