If they pile on after you have conceded, typically they come off much worse socially in my opinion.
In my experience, it's something of a sliding scale as you go higher in the amount of politicking your environment requires.
Lower-level engineers and people who operate based on facts appreciate you admitting you were incorrect in a discussion.
The higher you go, the more what matters is how you are perceived, and the perceived leverage gain of someone admitting or it being "proven" they were wrong in a high-stakes situation, not the facts of the situation.
This is part of why, in politics, the factual accuracy of someone's accusations may matter less than how successfully people can spin a story around them, even if the facts of the story are proven false later.
I'm not saying I like this, but you can see echoes of this play out every time you look at history and how people's reactions were more dictated by perception than the facts of the situation, even if the facts were readily available.
Admitting the mistake in front of strangers, usually leads to them making the shortcut next time and assuming you are wrong again.
you wont get any awards for admitting the mistake.
There is a reason why CEOs are usually a showman type.
Those who are obsessed with their reputation above morality have usually had a lot of practice punching down, and they don't really look as bad as someone being dunked on who gets confused.
I think this is like a cornerstone of bullying. It seems to work in social situations. I'm sure everyone reading this comment can picture it.