Push support has been there for a long time now. You can enable it on a per-folder basis, check in Manage folders.
Usually you rely on a server telling the mobile app that there's something new to poll (or the message is directly sent and displayed).
Also, I am just curious if you're unhappy with their android app? I find it very full featured
But no, it's a formal android app and I offer two pieces of supporting evidence: the resource browser shows the apk very obviously has a bazillion android resource files <https://imgur.com/a/pUbQ8Bp> and it supports native fingerprint auth
Native fingerprint auth can be implemented on top of a webapp; that isn't evidence either way.
Either way, no offline access in an email app... pretty bad.