Update:
I know uncovered boxes are recommended. I've tried tall and short uncovered boxes. Someone always went potty over the edge. I went to all covered boxes... the biggest I could find. No doors, and dust and perfume-y free litter. Hoping I was doing good, by not forcing the cats to use small enclosed areas that gagged them or caused respiratory issues. The sneezers stopped sneezing, so I assumed I made some good choices.
With this box, where the top meets the bottom... there is a latch. Someone has good enough aim, that s/he gets potty between the top and bottom...via the latch mechanism. This is a favorite box (no matter where it is located), soooooo I wash the potty off...life goes on.
This box bottom is taller, than any of the other covered or uncovered boxes, I use or have used.
I thought I would try something. I removed the lid last week. No one has gone potty, over the edge. There is no potty on the walls of the box at all! House walls are clean. Floor is clean.
Either the ones that are normally side-of the-box or over-the-top-shooters, aren't using the box.... OR since there is no lid, the shooter automatically squats. I am guessing it is a combination... height, width, and no lid.