This is a good point, but also a sensitive one. If we're not mindful of our emotions and end up dragging the conversation for too long, we could end up fueling the Continued Influence Effect - where any evidence that challenges people's beliefs only strengthens their inaccurate belief.
Personal experience has taught me that ending such conversations early, before people turn defensive, helps. It gives them the space to let a thought germinate in their own heads, and eventually accept it thinking it was their idea all along. (It's almost impossible for people with strong opinions to change their minds overnight. It would be a miracle if they change in a week.)
If people still refuse to admit it, I'm training myself to be okay with that. In the end, it comes down to what matters more: the relation, or being right. If it's the latter, it's simpler to walk away quietly and not engage with them again.
This comment is in good faith, not an attempt to prove anything. Cheers!