"If I back down, I'll look weak"

This is the belief underneath the others, so it goes first. You said a thing, you were wrong, and admitting it in front of people will lower you in their eyes. Confidence is part of how they see you, and a climb-down is confidence being withdrawn.

Adam Fetterman and Kai Sassenberg tested a version of this on people whose standing depends more than most on being right. In 2015 they surveyed 281 published scientists about what happens to a researcher's reputation when a finding of theirs fails to replicate, which is about as public a way of being wrong as an academic can manage. The scientists overestimated how much damage the failed replication would do. And when the researcher admitted the finding was probably wrong, the damage to their reputation was smaller than when they didn't. Admitting it wasn't the thing that hurt. Refusing to was.

That's one study of one profession, and a scenario study at that, so hold it lightly. But it fits a much larger picture of how people judge each other. Susan Fiske, Amy Cuddy and Peter Glick, reviewing the research in 2007, describe two dimensions on which people size each other up almost immediately, warmth and competence. Warmth is whether you mean well. Competence is whether you can do what you intend. The two are judged separately, and people who come across as high on both get the best of every reaction.

Now look at what a climb-down does on each. On competence, it costs you a little, because you were wrong about something, but the other person already knew that, and the amount they knew it didn't change when you said it. On warmth, it gains you a lot, because someone who can say "you're right" is someone who isn't going to fight you for the last word, and that's the first thing people want to know about anybody. Holding runs the other way. It gains you nothing on competence, since the wrongness is already visible, and it costs you on warmth, because now you're someone who'd rather be wrong than say so.

I'm extending Fiske's framework here rather than reporting a finding about climb-downs, so treat the accounting as my interpretation. But the direction is the point. The weakness you're worried about is a small loss on a dimension where the loss has already happened. The thing you're not counting is the gain on the dimension people actually watch first.