Oh I forgot to comment when I beat this. I learned some deep secrets of the universe about trapdoor mazes when I finally proved this possible.
'Trapdoor maze' is a reference to Deadly Rooms of Death (DRoD), where an entire archetype of puzzles is 'a bunch of tiles you can only step on once; step on all of them'. In this context, the darker blots are like multi-step trapdoors, and the logic for 'how to traverse them' is still deeply related.