1) Turn Puzzles OFF.
2) Enter Lexus' Tomb, 12-pillar puzzle room.
3) Drain the room of water.
4) The two doors will be open, allowing you to acquire the treasures inside.
5) To get the Star Key, refill the room with water.
6) The two doors will now be closed, and you can't open them with water in the room.
The doors are open when the water is drained, but not when the water fills the room again, thus forcing you to do the puzzle.