Guys, this wasn’t actually fixed earlier. My bad. But it is now!
The problem was with the mail server I use to process incoming replies, which runs on cPanel. By default, cPanel had two settings that needed changing. I can’t recall the first one right now, but the second was related to handling the “+” symbol in email addresses.
Previously, I was paying for a pop3 inbox from Hostinger, but like other mail providers, they stopped offering pop3. So I decided to try to self-host the service on an existing cPanel stacklinux.com VM.
But while cPanel supports “+” addressing, it was automatically placing those messages into subfolders. Because of that, the Discourse forum software couldn’t find the messages to relay and post to the forums. 
I’m writing this down here both as a reminder to myself for the future and to help anyone else who might run into the same issue.
This was the fix:
For reference, here’s how the reply by email works: