From what I've read in this and other threads, making a change -- any change! -- in the account's email settings can sometimes succeed in solving the problem. But sometimes not.

This is what I've done, so far:
* Deleted the account and re-created it with all default settings.
* Turned on and off various settings
* Turned on "retrieve messages and attachments"

But, still, all messages in the Inbox and Sent folder are gone when I leave and re-enter the Mail application.