An open staircase is likely to have either plaster, or drywall on the walls and ceiling. A hole in the ceiling between the joists in a floor/ceiling is a wood lined chimney and will rapidly spread a fire. Used to be you had to put 'fire blocks' in walls, a 2x4 across each vertical 16"o/c stud wall gap, so each vertical gap had some impediment to fire traveling up the wood sided chimney. I'm not sure if this is still code in most places or not. but was when I built my last house.
Some fire laws and codes are interesting, like 5/8" drywall required between a garage and a living space, fire blocks in stud wall passageways, and air gaps around chimneys, but the new 'zero clearance' chimneys make me pause and think a bit...