Many users, including myself, have a desktop with thunderbird basically always open in the background. It would be a very nice feature if the desktop client could act as an "Application Server" (as per the unified push docs) to relay to the mobile app (or maybe even another desktop client) that a new email/message has arrived and that the it should check for it.
Molly (a fork of Signal) has a similar feature to what I am proposing called mollysocket.
This feature would make it possible for the mobile app to conserve quite a bit of battery because instead of it always staying online and checking for new messages, it would only try and pull them when the unified push distributor receives a signal from the desktop client that a new message is available. It basically combines the speed of Push mode and the battery savings of Polling mode by shifting the load of the active connection to the desktop.