Gale (reference in the comments) famously showed that the determinacy of n-player, n-dimensional Hex is equivalent to the Brouwer fixed point theorem in n dimensions.

We can (and Gale does) view this as saying that if you d-color the vertices of a certain graph -- specifically, the graph with vertex set $[n]^d$ and two vertices $v, w$ adjacent iff the max norm of $v - w$ is 1 and all the nonzero components of $v - w$ have the same sign -- then there's a certain monochromatic path. Alternatively, you can think of d-coloring a d-dimensional $n \times \ldots \times n$ cube, and the determinacy of Hex/Brouwer fixed-point says that a certain "twisted path" must exist.

Here's what I want to know: Is there a topological proof of the density version of the determinacy of Hex? The density version ends up following from density Hales-Jewett, since combinatorial lines are paths in the underlying graph. But DHJ is hard, and this seems like it should admit a proof along the lines of Gale's.

What I mean by the "density version" is: for any $\delta > 0$, and fixed n, for sufficiently large dimension d any choice of $\delta n^d$ moves must connect two opposite sides of the hypercube/d-dimensional Hex board. (I'm fairly sure this is the correct statement, but it's possible I'm wrong. Let me know if this is for some reason utterly trivial or false.)