I'll try to bring it all together. Here is an overview so that you can see the connections:
https://www.logitechg.com/de-ch...
The adapter has USB-C, but that doesn't help with the PS4.
The headset only has a 4-pin jack plug. Normal for headphones would be 3-pin.
The PS4 DUALSHOCK 4 controller has a stereo headset (!) connection, not just a stereo headphone.
https://www.playstation.com/de-de...
Stereo headset connector must be 4-pin... if that's what it says.