Another weird "what if" scenario for Army/Navy

Suppose they both go undefeated and play in their conference championship game a week before the "real" Army/Navy game (horrible for Notre Dame although great for college football, but I digress). The winner is destined for the college football playoff.

But what if the winner of the playoff game turns around and loses the next week? I researched that their traditional game is actually a non-conference game.

I would guess the winner of the conference title game is still in as the playoff field is set at that point. But what a potentially strange occurrence.