Note 7 runs Android 6.
And Android 6 has this thing called "Doze Mode" where -- exactly as you've described -- the phone will go into a special "battery saving" mode after a few hours and will severely limit what apps are allowed to do in the background.
Google's official workaround is -- "hey developers, use Google's Cloud and move your user's messages there" -- which doesn't work for me (or my users, I suppose).
Now, in the current "beta" version, I tried to make things better, but not sure how much it helps if any.
You can install it from this link:
https://play.google.com/apps/testing/org.kman.AquaMailOh and one other thing. On an Android 6 device, AquaMail needs to be excluded from this "Doze Mode".
Please check app settings -> mail, receiving. If not already excluded, there will be a panel / button to do so.
( this does not unfortunately fully revert to how things worked prior to Android 6 )