Safe to open up a main floor stairwell?

I have a typical 1920 Toronto semi where the stairwells leading to the basement and second floor are against the inside neighbour adjoining wall, one above the other. The house is only 15.33 feet wide, and single uninterrupted beams run the width of the house except forthe stairwell area.

I want to open up the main floor to second floor stair well - just 3/4 way up the stairwell and will not be opening up the basement stairwell. From what I can tell this is not a load bearing wall. Below it, it's open in the basement with a thin piece of wood (not even a 2X4) at the base of the railing.

The wall I want to partially remove is approximately 4" wide. That wall does not continue on the second floor to the ceiling; there is an open rail that is original to the house. Is it possible that this interior stairwell wall helps support the main second floor however? Since there isn't a wall on the second floor, and only a thin 1" piece of wood connecting (sort of) the basement stair handrail to a basement floor joist, safe to assume this cannot be a load bearing wall?

My neighbours with exactly the same house as mine have opened theirs up however they are not the sort to get permits or have engineer checks. All of these houses have already been opened up from formerly tiny rooms on the main floor.

Thanks for any advice!

D