Oyeve
Lifer
I would be surprised if the adapter had a built-in DAC. The odds are that the phone has a DAC built-in, and the quality would be the same as if you had a native 3.5mm jack.
No way that they would sound equal to plugged in cans. Bt headphones stink in general wireless. I have a few that work wired or wirelessly and always sound way better wired.