"I don't want to give them the satisfaction"

You can see it on their face already. The small smile, the "told you", the way they'll say "so I was right" at the in-laws. The climb-down hands that to them, and there's a part of you that would rather sit in the roadworks for an hour than see it.

This one deserves a straight answer rather than a study, because there isn't a study that reaches it. The gloat is real. Some people do it. And the belief that holding denies them the satisfaction is wrong in a way that's worth spelling out, because holding gives them more.

Watch what the gloat feeds on. It feeds on a contest. "So I was right" is a sentence about a competition, and it lands when there's been one. If you've held for ten minutes and then, finally, conceded in the roadworks, the gloat has a lot to work with. There was a fight, they won it, and the concession was dragged out of you, which is the most satisfying kind. If you say the five words at the turn, twenty seconds in, there's been no contest. "The third, you're right, I forgot the roadworks." What does "told you" land on? There was nothing to tell. You've given them a fact, not a victory, and the gloat, if it comes, looks odd, because it's a victory lap round an empty track.

Then say nothing. This is the part that takes practice. The gloat wants a response, a "yes, all right" or a "you don't have to be like that," because the response is what turns it into a second argument. Give it nothing and it dies in a sentence. Most people, hearing themselves say "told you" into a silence, hear how it sounds and stop.

There's also a longer answer. The gloat is theirs. It's a thing they do, and it says something about them, and the people watching can see it as clearly as they can see your climb-down. You're not responsible for how they take your honesty. You're responsible for the honesty. If someone gloats every time you're straight with them, that's information about the relationship, and chapter 24 comes back to it. But the number of people who gloat at a clean climb-down is much smaller than the number who gloat at a dragged-out one, because the clean one gives them nothing to hold.