If you don't use GCM / push - it depends.
Our push code tries to deal with restrictions of recent Android versions - but a lot depends on how lucky *you* are (the specifics of your mail server and network connectivity).
We use GCM push (Firebase Messaging to be more exact) for Exchange accounts and it works well, without overhead or privacy issues.
But for IMAP - we'd have to make our app "cloud based" - collecting user messages into our cloud and pushing changes from there - and the cloud would have to have access to user's accounts (and messages). We don't have any plans for this.