Oh for fuck sake...reallyIts doable with very accurate jumping. Start on the small box in the bottom corner, then the box next to it, then a long jump to the stack in the far bottom corner. Then, jump from the highest box of that stack to the small box in the middle of the room, then to the top left corner stack, then just hop up onto the mezzanine.
A tip; you can leave the long jump timing later than you'd expect without falling off the box, thanks to collision detection and sprite sizes.
I was struggling with that first big jump on the bottom, then i followed your tip advice. How to say this...i used the jump when i was about halfway between those boxes.