In this lecture on the Matchmaker Game, the professor says that the payoffs from the pure strategy Nash Equilibria of the Battle of the Sexes subgame can be rolled back via backward induction to the matchmaker's decision node, because the matchmaker believes that the players are going to play a Nash Equilibrium in the subgame.
Why is this reasonable? Isn't the point of Battle of the Sexes that even rational players cannot be guaranteed to play such a Nash Equilibrium? They may fail to coordinate even though they are trying to do so. I know there is a Burned Battle of the Sexes version if one of the players can signal to the other that they have the ability to burn money, which does have guaranteed coordination, but this has not been mentioned in the course. I do not see why the matchmaker should form a belief that the players will reach Nash Equilibrium.
He does also mention the mixed strategy Nash Equilibrium, but this is a separate analysis.