Rogersia is not likely at all. Looking at various websites, Rogersia is a shade plant (or a sun plant if it gets moisture), and it would not be able to grow in a hot dry hole in a sidewalk next to a tree stump. And Rogersia leaves apparently grow individually, each on their own leaf stalk that comes out of the ground, or arranged alternately on flower stalks. This plant has leaves arranged oppositely on stems, so it can't be Rogersia.
Aesculus has opposite leaves, so it is a definite possibility. Since the stems appear to be sprouting directly from the base of the stump or the roots around the stump, they're clearly young stems of a tree, and so horsechestnut or buckeye are likely. As for the leaves not looking exactly like mature horsechestnut or buckeye leaves, that's understandable since it's just a sprout that hasn't fully unfolded its leaves yet.
