An update on this:
I switched from “Always check for new comments and update automatically” to “Always check for new comments and show update buttons”.
The update buttons do appear as expected, such as “1 new reply on your comment” and “2 new comments”. However, when I click the button, I see the animated status bar thing, but the comments don’t load.
Oh, and I’m using wpDiscuz – Supercharged native comments version 3.2.5.