Can I use my XBox controller as wireless headphones?

I have the new Xbox Series X controller and am wondering (as I have a very short headphone cable) whether I can connect the AUX port on my controller to my headphones, and then have an audio connection to my PC via Bluetooth.

Aka I plug my AUX cable into the Xbox controller that is connected to my PC (Windows 10) via Bluetooth. How can I use it to hear my PC audio?

This only works with the wireless adapter, not with Bluetooth.

Then it works fine with plug and play.