That’s exactly the reason why I (with individuals I believe to require it) try to point out a tiny detail that still somewhat sound like it formerly supported me while admitting it, or explaining in great detail what got me bamboozled. Just to distract the other person, which I do not trust to not immediately use my mistake against me, to such a degree they can’t do it. Alternatively swing it like “ooh, this is like when you got XY wrong!” so the playing field is somewhat even already before the other realises you just admitted something.

It sucks, but it’s better than not admitting it at all. Fortunately not a lot of people I know in my life require this.