The fact that is only INBOX makes it sound like a pop client is fetching the mail and deleting it. The logs should show access.
If it were other folders as well, I'd suggest checking zimbraMailMessageLifetime. Probably still worth checking. Run:
As long as you see something like that then it isn't message lifetime. If zimbraMailMessageLifetime is set to 1d then that is your problem
/opt/zimbra/bin/zmprov ga your-account|grep -i zimbramail.*lifetime