I wasted a couple hours trying to help one of my clients who had this problem last week. What a pain it was. He had 5 devices connected to his account, 2 on pop and 3 on imap. It took 20 minutes just so listen to his explanation of emails appearing and disappearing and re-appearing based on which account saw them first. We spent the next hour turning on each device one at a time until we determined that his outlook pop client was the one deleting them, even though it was configured to leave them.