How to solve rat riddle in Shido's palace?

I'm trying to get my second letter in Shido's palace. At the restaurant, I took an elevator to Mid Starboard Hallway.

Here, one turns into a rat within grey rooms with activated statues.

I know that one as a rat can go through vents (small, white paths) and that there's a switch in one room ( yellow check mark) that turns them off.

Yet since all the doors are locked (yellow lock symbols) I cannot proceed to the other areas (all in red).

I'm stuck here, going in circles ad nauseam and am unsure how to open those locked doors.

Can I find the solution here or do I have to go somewhere else first? How to get the doors unlocked.