The point of a mail server is that many clients can connect to it and get what they need. What I’m reading here is that you want to disregard the ability to already do that in favor of having all your mail funneled through a self-hosted ‘something’ that just sends push notifications to your mobile…but then you’d still have to have your mobile mail client go and download all this mail you said is a battery drain, so you’re sort of negating yourself.
Now…the real crux of the problem you’re describing is simply that your mobile mail client is not very efficient, so why wouldn’t you just solve for that instead? Create a better workflow for your mail so your client doesn’t need to IMAP crawl EVERYTHING, or reduce the frequency it syncs maybe.
If that’s still not enough, depending on your mail host (which you didn’t mention), there are ways to simply subscribe to push notifications from their service more than likely if that’s all you want.
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