To process all **frozen** messages in the Exim queue use this command:

mailq | grep frozen | awk '{print $3}' | xargs exim -v -M


I believe that I have fixed the problem on my own, but it may still be a bug with cPanel.

I found the following file:


Deleting this file fixed the problem immediately. I'm guessing that the system should have deleted (or updated?) this file at some point, but didn't.