Ever had a stupid conflict in public? You know you're right, but you will never convince the other person, because he will lose face. And the details are not the point anyway.
There is another way:
"I didn't know that. I was totally wrong, and I apologize. Say, those earrings are fantastic! Did you make them? They really make your eyes light up. Where might I get something like that for my niece?"
It seems like this approach (shortened form) would be so fake that anybody can see through it. Many can, but it WORKS ANYWAY. It does not even have to be a compliment. Simple talk in a friendly way about something the other person is interested in, with sincere interest, and within one or two minutes at most it will be almost impossible to for them to stay upset.
I know it works because it was used on me. A person came over and ordered me to come to a meeting with a salesperson. I pointed out she could not give me orders because I was not her junior. Very clumsily with no segue she started talking about the photo mag I was reading, and in half a minute I had no energy or interest to be upset, and accepted her invitation (no longer an order). So even when very crudely done, and on a person who is fully aware what is going on, it works.
By the way, don't worry about being a fake. You can believe anything you want, can change any belief anytime, and there's enough truth in anything for somebody to believe it.