Bard’s song of rest ability also currently resets when you switch regions, like Rivington to Lower City… giving you infinite short rests if you want it
Never really tested it myself, so cant confirm 100%, but you technically aren't changing regions when you travel to and from the underdark. There is no loading screen. It's all part of the same map, I'd assume.
It might work with the Mountain Pass, though. But I'm not sure.