When the two reach the front door leading to the study room, Minzhe excuse himself. He felt that it wasn't his right to be there during their confrontation. Especially that the two might have secrets that they need to talk about, he might get kick out still by his father. Leaving now would end up the same thing.
As King Jin opened the door without even announcing himself. If the person inside was a part of himself, they would be able to have a consensus about how to go through this motion.
