There is no staircase, the dev messed up there imo. Think they were suppose to add some way to jump down to the chest place, and a way to return, but forgot. -- The chest is in the spot where the stairs should be.
Here's the data file, where I modified the maps a bit so you can proceed. [Don't remember the map name/number, so it's the full folder.]
One of the tiles allow you to "jump" down to the other chest. After you can then "jump" back to exploring. -- Was lazy, and not my game, so it's just a simple transfer event.
Also should semi-fix the examination tower--added a 'switch' to let you open the portals. [They really should have used self-switches here; remembered to reset the switches after you leave/use the dark portal; or evented it better, such as used variables or more switches.]
Edit: Added the complete game save; not sure if you need anything besides the save file, but if no images shows, just talk to the fairy on the left.
Edit2: NVM, fairy seems to always be there, so you don't even needa download any save; just play the game till you get the dimensional box, use it, travel left & talk to the fairy.