Yes, I would say if there are multiple playable characters, they all need to have the false belief. Otherwise, why would the others who don't believe it keep working toward the goal? If they know they are operating under a false premise?
That's what I'm struggling with. By the guidelines, the player DOES know they are working under a false premise. The character believes something the player knows to be false, yet in the shoes of the character is required to play along. It's a terrible dissonance.
And it's entirely possible for multiple playable characters to have concurrent or even opposing goals...