Unless I'm misunderstanding, with this advice you run the risk of shutting down some converstaions which may be important to one person, but not particularly to the other.

We should probably distinguish conversations that are about just something in general, and examples in the post, which are about potentially crucial/difficult challenges a person is having.

I may not be particularly interested in a problematic situation that someone may be having at work, but I know that by listening and showing genuine curiosity, I may be able to help them in their own understandin and reaching a potential positive outcome.

What if my own interests aren't an option?

Ask yourself why you want to be a better listener when you don’t have any actual interest in what they have to say.