Well, I've knocked off Mike Tyson, Mecha Birdo, Mother Brain, and the Clown Copter, and also gotten through what there is of the unfinished path, but I can't even begin to figure out where the fifth completed boss is hiding. Here's what I've done so far:
Beaten every boss except Mother Brain at least twice. (Mother Brain's path closes off after you beat her, so I think we can safely assume that the fifth boss is not along that route.)
Mapped out every screen I've found. It's not just a two-dimensional map; there are some rooms that would overlap if everything fit together like you might expect.
Combed several rooms for invisible blocks: The two rooms on the way to the Game Over room, the Zelda room, the Zelda cave room, and the room right after Mike Tyson.
Pushed at every open-looking wall I can reach.
Tried to jump off the ceiling of the room before Mike Tyson. You can get up there by jumping from the second moving platform onto the top of the second block with a spike coming from the bottom, but the top of the screen seems solid.
Got Gutsman to hit those mysterious spikes that fall on you if you double jump in the right spot in that room.
At this point, I'm beginning to seriously question whether this alleged boss actually exists, or whether the assertion on the website is simply a cruel lie, in keeping with the theme of the game. Has anyone found anything anywhere that even hints at the existence of another path somewhere?