This may seem obvious, but are you absolutely sure that the statue plans aren't already on the chalkboard? Many people have mistaken one for the other, and we have had no reports of problems with the bath plans getting "stuck." I have repeatedly tested this part of the game, and have seen the plans work perfectly every time (regardless of whether puzzle 12 is complete or not).

If you have level 3 in both restoration and leadership, what happens when you drop a builder on the path near the statue?