Elsholtz–Rackham counting conjecture for higher-dimensional grids

Let $[n]=\{1,\dots,n\}$, let $[n]^d$ carry coordinatewise addition, and let $k_d$ be the maximal volume of a region in the unit cube $[0,1]^d$ bounded between a hyperplane $H$ and its dilate $2\cdot H$.

Elsholtz–Rackham's higher-dimensional conjecture. The number of sum-free subsets of $[n]^d$ is

$$2^{k_dn^d+O_d(n^{d-1})}.$$

This extends the two-dimensional counting conjecture using the extremal constant $k_d$. The supplied text does not state a resolution, so the conjecture remains open.