The rules as written are not ambiguous.
Round is defined as once around the table, 1 turn per player.
Palace Royal is played in rounds. The start player begins and executes his complete turn. Then, the next player in clockwise order takes his turn, and so on.
The card in question:
The player receives the majority bonuses in all spaces where he has at least one servant. He keep the card face up in his play area until the end of the round.
So, technically it should stay out the entire round.
But that's not really relevant anyway, because other players still get their majorities regardless if the card is in play or not.
Note the wording on the card - "receives the majority bonuses", Not "has the majority". IE, you get the majority bonus, but officially you still have a minority there. So when the actual player who has the majority plays his turn, he also gets the majority bonus.