I've been e-mailing with the support team. Léo asked "Can you let me know if the track in question has a chain enabled on the scene you've selected?"
I finally was able to check, and I did. Now - I had a chain saved , but I did not have a chain enabled.
But, good news! Erasing the Scene did make the behavior go away! So, it's not a great workaround, but it's a workaround!
I do believe this is a bug, based on not having the chain selected/enabled. And, as far as I can tell, the only way to erase a chain is to erase the entire scene. But, that's OK - I can work with that. It lets me use the looper again, anyway! I hope they're able to track down this bug.
Can other folks verify this works for them too?