I realize this is much more academic than something you are going to have to do more than a couple of times in a career. You can prove to yourself that this gets close. Tape measure from the the bottom to the top of a file cabinet so you know the difference. We know that even on really calm days, the barometer is going to drift and this must be accounted for so one set of readings on a gusty day is not much use. Set the device on the floor at one point and wait long enough for it get to ambient temperature, say 3 minutes. Then take 10 readings at 10 second intervals and average them. Then go to the other end of the stairs and do the same thing. Repeat this a couple of times more and see what the differences are. We know that these instruments are very sensitive. You can see the change just moving it from your belt to your chest. It is just a matter of lots of tedious readings to get the averages to work for you.
My opinions is that this is the best you are going to do short of hiring a survey crew.