That wall in front of it can be broken, use the horse jump on those lamps to broke is the leave the area and go back to the lamps re-spawn and use them to get there.
So i just did it, i don't know what the horse jump is though, broke the wall with the charged dog attack from the other side, and then slid through, timed the jump (probably frame/pixel perfectly) to jump up to the axe, I only have the star and the dog upgrades. Did i kinda sequence break this one?
Edit: (also, died just after i got it and before i could get to a save, gonna try and get it again.)
Edit: addition to last edit:
So I managed to do it again, once you know how to did it, it's kinda easy to do, just jump at the last bit in the tube and hold left. you'll get it like 80% of the time if there's not damaging debris falling there.