Credit goes to @Cytruss for this discovery. The idea is that you can skip levels but pressing "next level" very quickly. With further testing, I discovered you can basically do it excessively, such that you can unlock up to the last level of the game. Skipped levels cannot be unlocked, and the glitch doesn't seem to work between unlocked levels. So (I think) you need to clear your game data before executing this. Finally, it's very easy to cause the game to crash and miss the final level. So this glitch actually takes some skill and good timing. I'll be making a glitchless category and moving all the other Any% runs to that.
Edit: @Cytruss had all levels unlocked beforehand, so no need to clear save data.
Here is the glitch, as demonstrated by @Cytruss:
