Welcome to the human race! Everybody wants to be right and avoid being wrong. Everybody wants to look good and avoid looking bad. We go to such lengths in that pursuit, and tangle ourselves up in the struggle.
But what's the pain of being wrong, really? If I see that I'm wrong, that gives me a golden opportunity to learn something, doesn't it? It gives me the opportunity unfold something in myself that I've been clutching. Admitting a mistake allows in the flow of something that works better. Plus, it really makes people happy when I admit a mistake! What a generous thing I can do for people!
So what are they gonna do, take away my children? Shoot me in the kneecaps? I guess there are mistakes I could make that would lead to those outcomes, but there's no real imminent danger in seeing or admitting that I'm wrong or mistaken. Emotional pain that would result is all self-created, which reminds me that I also have the power to acknowledge and release that emotional pain. And then I would be free to reap all the benefits of being fully self-expressed, and cleaning up as I go along. I think it would work for me to focus on generating love and affinity in the conversation, rather than worrying about stepping on a minefield. What would work for you?
And of course I'm probably wrong about all of this!