I do not think that we can expect a topological proof. Note that the topological proof of the coloring theorem does not rely on assuming n being fixed and d is large. when d is fixed (two) and n is large the coloring result is correct but the density result is far from being correct. When n is fixed and d is large the nature of the problem becomes similar to density Hales Jewett (while weaker) since (#)the paths in the graphs are not so different from combinatorial lines(#).
(Of course, if the question is "Is there a topological proof in the literature then I am quite certain thet that answer is no.)
The problem is interesting as it is a problem of a type "can we expect a method X leads to a solution of problem Y". Definite answers and even formal ways to pose the questions appear to be very hard. On the other hand, some heuristic arguments can be useful. But I am not aware of many such questions/heuristic answers in the literature.
Regarding my heuristic argument, the statement between the two (#)s looks especially questionable. But even if it is not the case that paths in the graph (when n is fixed and d very large) are "similar" to combinatorial lines, I would still regard the possibility of a topological proof rather unlikely.