People rarely change their minds mid-argument. They change them in the shower three days later. Give them time.

Nobody concedes while being watched. In the middle of a disagreement there is too much at stake socially, so people defend a position they may already be unsure about, and the actual reconsidering happens alone and later. Winning the exchange and changing somebody’s mind are almost unrelated activities.

Make your case once, clearly, and then stop pushing. Leave them an easy way to come round without having to admit anything. Check back in a week rather than demanding an answer in the room.