I heard back from Apple support this afternoon with this message, which is a reply to my request for more information:
The best information you can offer to GoDaddy is the logs - the logging will show the GoDaddy engineers what is going on, and that should be the data they need to help analyze and resolve the issue. You can get the logs from Console application.
I replied with the following:
Thanks, but as I suspected, GoDaddy has declared that it’s an Apple mail problem, since they only see it (and they’ve been seeing it a lot) with El Capitan. It turns out that IMAP between El Capitan mail and GoDaddy IMAP mail simply doesn’t work (BTW: I’ve see the problem with my other GD IMAP mailboxes). The problem is much more wide spread than I originally thought.
In some of the Apple community forum threads, there seems to be some acknowledgment that it’s a problem with the agent and will be addressed in either the 10.11.3 or 10.11.4 point releases. Since I don’t know the players, I can’t vouch for the accuracy of the comments.
But at this juncture, both companies are pointing to the other, which is very poor customer support, especially for a critical feature like email. In my opinion, there was a change to IMAP support in the agent that is either the problem or exposed the problem. In addition, there is likely an issue with how the protocol is handled on the GoDaddy side, since we are not seeing this with other IMAP services (that I’m aware of). This will only be fixed when both sides work the issue or at the least, Apple puts in some defensive code.