My guess would be not. That'd probably decrease their applications, which already dropped a fair bit last year. Then again, if they're looking to increase yield, they very well might add the essay. I suppose you'll find out quite soon.
FWIW, the Dartmouth supplement was one of the very last pieces of my Common App that got submitted (something like a day before the deadline). I have no idea why I waited so long - I don't think I even saw the thing that requested you submit it early - and it didn't matter in the end. Dartmouth accepted me, while I was waitlisted at other schools in which I'd shown a LOT of interest. It's all just a game, ain't it?
@jnix, you are correct in saying that the essay is an important part of any college application, but OP was talking about a supplemental essay, not the main (Common App) essay.