The word "Converse" can certainly be used as a noun in other applications of English, but saying anything like that a participant in a conversation "had converse" is surely out of the question?
When you converse, you are having a conversation, or have a conversation (participation implies ownership), and afterward, you have conversed, and have had a conversation. There is - to me - no conceivable way to have or have had a "converse". Or, e.g. "your converse is heard in conversation". This can only be when you are discussing noun converses.