If it fails at that then it is a pretty significant problem. As you say earlier "the refutations are in the training data too", then the LLM should in fact be able to use "both sides" and land with a little better confidence when presented with new data.
(Hopefully your point regarding prompting issues is resolved then.)
I was just refuting your contention that this is somehow inherent in the idea of "training", and it's not.