windows 10 – Can I get the audio of the XBox Wireless Controller X Series to work when connected via Bluetooth?

I have a a Wireless XBox Controller X Series connected via Bluetooth (no wireless dongle) to my Win10 PC. I can play games fine, yet I cannot see a new audio device.

Connecting the audio jacket with a headphone won’t play any audio when on Bluetooth.

If I connect it via USB-C the sounds plays fine and the audio device is discovered as Headphones XBox Controller.

Is there something I can do to get sound being played through the jack? Is my device broken or is it not supposed to work via Bluetooth connection?