Aaronson's lattice variant of the Sensitivity Conjecture

Let $C$ be a two-coloring of the $d$-dimensional lattice $\mathbb{Z}^d$. It is non-trivial if the origin is red and there is at least one blue point on each axis. For a point $x$, let $s(C,x)$ be the number of neighbors of $x$ with the opposite color, and let $s(C)$ be the maximum of $s(C,x)$ over all points.

Aaronson's lattice variant. There exist constants $c$ and $k$ such that for every non-trivial coloring $C$,

$$d \leq c\cdot s(C)^k.$$

This variant asserts that the dimension and lattice sensitivity are polynomially related, and Aaronson showed that it implies the Boolean Sensitivity Conjecture. The paper studies reductions and constructions relevant to this conjecture, but does not report a resolution.