How is the audio compression codec[0] negotiated between the phone and the headphones over Bluetooth? IIRC, Sony supports higher quality codes outside of the standard BT required ones. Is the app required for that negotiation or is it all in the operating system now?
[0] There is no lossless high quality audio over BT, only a bunch of lossy codecs.
IIRC, the app isn't actively involved in bluetooth audio negotiations, but it does allow you to change settings within the headphones around what codecs it will advertise support for and prefer to use. Those settings have reasonable defaults and any changes you make persist on the headphones even if you uninstall the app.
[0] There is no lossless high quality audio over BT, only a bunch of lossy codecs.