I have a backup client on my computer that automatically sends me emails for each completed backup task each day. On Gmail I have a filter set up to make these emails skip my inbox (straight to archive) so they don't pollute my inbox. This works great on the Gmail web client, but the iOS Mail app still shows notifications for these emails, even though they don't get put in my inbox. If I check my inbox on the mail app, the emails don't appear either.
How can I make iOS stop popping up notifications for automatically archived emails, but still get my normal notifications for everything else?