I would recommend going and watching the pannenkoek video because it is so good.
The quick version is that it has been known to be possible to clip through the wall to get to the door, but Mario has to be in the walking state to open it. Since the floor directly under the door hitbox is all covered by wall, and the part that isn't covered by wall doesn't have a floor directly below it, it seems like there's no way to actually be in the walking state to open it.
However, by abusing an exploit where Mario finishes his turn around animation the same frame he leaves the ground, the transition to the freefall state is overwritten by the transition to the walking state; this lasts one frame until a check is run next frame — which properly puts him in freefall. Timing the turn around right lets you be past the wall and walking, so you can open the door.