No, because i'd end up trapping people besides smith down there. I'll just take my full movement for that move action and end up in A4 on the second floor then, leaving my at the landing.
Just a headsup team, on my turn, I'm burning the stairs. If they drop, I'm fivefoot stepping to A5 then shutting the door.
Edit: correction, John is still in A2 on my turn, so I'll Ready to burn when he passes me...