Mail disappearing on MacBook (att.net email on Apple Mail program)

I am visiting my parents-in-law and trying to assist them with computer issues.  My MIL has a MacBook as well as a very old Mac desktop.  She has had the MacBook for a couple of years and gradually moved to using it for almost everything she does, but has noticed that emails "disappear" from the MacBook after a week or two, but were remaining on the old Mac.  I checked settings and discovered that the old Mac has email set up as POP but the newer MacBook is IMAP, which explains why the emails are not disappearing from the old one.  But what we really need to solve is why the emails ARE disappearing from the server.  The inbox on the MacBook is the same as the inbox on webmail ... today the oldest message there is dated 12/24.  A couple of days ago when we checked, the oldest email was 12/23, so one or more messages disappeared in the last few days.  I do not see any settings that appear to automatically delete inbox emails, nor any filters or similar.  (I am a Windows user, so I fumble around on Macs.)

Apple customer support says call AT&T and AT&T customer support thinks it is an Apple problem. (surprise, surprise!)

Any ideas or suggestions?  


Are the still checking email on the old mac?  If they're logging into the pop account - all new mail will be downloaded on the old computer.  Are they not noticing new mail showing up there?


On the Mail > Accounts window, there's an option to "Remove copy from server after retrieving a message" - Immediately | After one day | After one week | After one month. This can be unchecked so that messages are never removed from the server.



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