The palace dungeon area is built quite deeply under the main palace. It can only be accessed via a hidden door behind the throne room. The only entrance brings you in front of a circular set of winding stairs that looked like it can go downwards forever.
Descending through it really made me dizzy but I couldn't use teleportation straight to the bottom since I haven't been to this area yet. The palace map I made doesn't include this hidden area. It took me almost 20 minutes to reach the lowest floor of the dungeon.
As I was moving down the stairs, I saw that the palace dungeon consists of a total of three floors which matches the report of the initial scouts sent in this kingdom. As per the briefing I had at Noir, the lower the floor is, the more important or dangerous are the people being kept in the prison cells. Of course, I would pick going to the lowest floor in a heartbeat.