So, we managed to get the pin out by disassembling the connector. It's really well glued in and there are lots of rubber bands, so you have to use some heater and pray that you don't break it. I can send more information, even some photos from the process if needed.
However, during the process the heater desoldered the wires from connector, so we have no idea where any of the wires should go.
Does anyone know how to wire it?
Lets use this schema for pin-number reference:
Since the headset has only 6 wires, this is how to connector looks: