Of course I haven't, I'd install microG if anything (but haven't yet). I don't use web notifications at all though so I haven't even thought about testing that - but as far as I know, Anbox doesn't have any way to push notifications out of the Android container into host system so far (but it shouldn't be hard to implement; plus I guess you could even just install KDE Connect inside Anbox and synchronize notifications that way as a lazy hack).

Thanks, do you receive notifications at all when in background in PureOS? Say for email?

Sure, it's just the regular desktop libnotify stuff.