IMAP Online allows to access the Sent Items but does not download locally the body of the email (it only downloads the sender, date and subject).
Any mail sent using a IMAP Offline account will cause a disparity with the sent mail on the IMAP server. This idea covers the request to incorporate this feature of an online IMAP account.
"Receives mail from the Internet server using IMAP. Copies mail messages from the Inbox on the IMAP server and stores them in the Inbox of Notes Mail (either local or server) listed in your Location document. You can either leave the original mail on the IMAP server, or automatically delete it from the server when Notes stores it locally."
"Provides direct access to your mail on the IMAP server without storing any mail locally. Uses a local proxy application to display mail and let you work with your mail as if it were Notes mail."