Some arguments need facts; others need both people to feel safe enough to stop performing.

Some arguments really are about facts and can be settled by finding out. Many are not, and no amount of evidence helps because both people are managing how they appear rather than saying what is true. Until the performing stops, the real subject never comes up.

Work out which kind you are in. If it is the second, safety comes before content. Lower the stakes, say something honest and slightly unflattering about yourself, and see whether the conversation changes character.