There is a hint in the other room that's opened by the sword. Worst case, just disable the puzzle and move forward!
There was the hint about all the oil being dimmer near the rear entrance, don't leave the keys downstairs ( assumed this was about the room prior) and something about cost of ladders and 607 and some other number being important. I don't see any numbers, there are no ladders, and when I find a dimmer room I can't find anything to click on and if I try it gets me to the next room.
I skipped it and realized how you did it and an arrow appears in a room where it wasn't before. I don't see how that is a clue. I mean i guess the crossed arrows maybe? But that's a huge stretch
But if Rumi has already changed the oil, and he's telling her to change the oil near the rear entrance it would mean that the rear entrance is where you want to go so you head toward the dimmer lamps.
I guess you define rear entrance as where they are coming from where I saw it as where they are trying to get to.