The hypercube maximum-density conjecture for monochromatic k-in-a-row

Let $[k]^d$ be the $d$-dimensional grid, with $k\in\mathbb{N}_{\geq 2}$ and $d\in\mathbb{N}$, and let $D(k,[k]^d)$ denote the maximum density of a configuration avoiding the relevant $k$-in-a-row patterns. Hypercube density conjecture. For every $k\in\mathbb{N}_{\geq 2}$ and $d\in\mathbb{N}$,

$$D(k,[k]^d)=1-\frac{2}{k}.$$

The source motivates this by a construction attaining the value when $k/2$ is a natural number coprime with $d!$; the assertion for all $k$ and $d$ remains open.