Sometimes, being honest means updating in the ‘wrong’ direction. Although, in the long term, you should expect your beliefs to drift toward the correct answer, there’s no reason to expect that you approach that answer monotonically (always moving in the same direction).
We can have better arguments if both players understand this fact, so no one treats it as a humiliating defeat if you concede that a piece of evidence meshes better with your opponent’s position than your own. That’s the expected sometimes, no matter who is right. Concede the point and then explain why it’s not enough evidence to make you drift across a critical value where your beliefs and actions change. There’s nothing wrong with saying, “That’s not the kind of proof I’d need. It favors your side, but my prior expectation that you’re wrong (for these reasons…) is strong enough that it doesn’t change my expectations very much.”
Talking this way lowers the stakes of each new piece of evidence, so we can consider them reasonably instead of treating any revision of our estimates as a coup de grace.