This Google guide talks about configuring ones Gmail (e.g.
firstname.lastname@example.org) to fetch emails from another mail service (e.g.
email@example.com) via POP3 into as well as being able to send emails as
firstname.lastname@example.org from Gmail.
But if you delete/move an email from within Gmail that was originally pulled over POP3, will Gmail also delete that message from the original
email@example.com mailbox, thereby syncing it?