I have Aqua Mail on a Nexus 5, one Gmail account set to IMAP Push. The problem is it's very inconsistent.
I have the Notification icon saying "Push mail: <my mail>", when I send myself a test mail, it changes into "Synchronizing with: <my mail>", and gets stuck there until I open the app, and it will refresh and find the mail only then.
However, if I just restarted the phone, mails will automatically be pushed, without getting stuck on "Synchronizing" phase, but only until I open the app and close it by sweeping it out. After that it behaves like I stated above, getting stuck.
(I have data sync enabled - does it even matter, I set Aqua Mail to ignore it and always sync. Do I need to have set "Scheduled checks" even while I'm using IMAP Push? Tried it both on and off either way, no difference in the issue)