With version 1.40.1 we have fixed the problems with the "Push" for O365 and exchange accounts.
In order to do that, please do the following.
1. Go to Aqua Mail Settings>Manage accounts>Choose the account/s>Folders>Disable the push for all the folders.
2. Aqua Mail Settings>Manage accounts>Choose the account/s>Options>Account options>Disable push
3. Aqua Mail Settings>Mail receiving> Disable push
4. Restart the app (completely)
5. Repeat the steps above, but this time activate the Push (no need to follow the exact order)