I'll bet if I returned in five years, you two would still be arguing about stalemate, each making the same argument over and over to no avail. At some point rational people would simply agree to disagree and get on with their lives.
If their goal is to convince the other then I think they are probably in stalemate. So, according to one of them, whomever has the last word wins, because the other forfeits by not responding. The reason for not responding is probably irrelevant, even if forced or if neither have sufficient convincing material.
But if the discussion ended in stalemate, it will go on, as they will again discuss about the stalemate rule in discussions.
If not that at least a draw by repetition. I'm sure the 50 move rule has also kicked in by now.