Going to the pause menu [p] and choosing Reset Cart(ridge) seems to fix the stair glitch. I've had it several times without even getting to 91. XD
Also progress seems to get autosaved, so even if you've left the webpage/closed the downloaded game, you should theoretically still be able to continue from your 91 run, you crazy lucky person. Incidentally that also seems to fix the stair glitch, but it's less convenient than just doing Reset Cartridge.