To be frank, if everybody else is sheltering in place, that means the staircases will be clear for me to run down. If everybody thinks as I do, maybe there will be problems. But, judging by my experience with staircases in America (namely them being fairly wide and numerous), I prefer my odds in the staircase. Even if I only manage to get a few floors down before becoming tangled in other humans, maybe those few floors are enough to put me below the fire, or below the floors now filled with toxic smoke.
I would not enter a stairwell that was filled with smoke. Smoke rises, so if there is smoke in the stairwell below me, it's probably also in the stairwell on my floor as well, though there may be a small window where I get mislead. Nevertheless, in America these stairwells are designed specifically to be used for evacuations, and in my city the official advice is to GTFO as quickly as you can using a stairwell the moment you hear the (building wide) alarm or if you see smoke. So this is certainly what I would do.
Having been a participant in the attempted evacuation of a 15 story building during multiple fire drills, I'm not looking forward to ever having to evacuate for real. After 2 hours under ideal conditions, the stairwells were still clogged with people, and nobody above around the 10th floor got out.
