The strict density bound for monochromatic k-in-a-row on the integer grid

Let $D(k,\mathbb{Z}^2)$ denote the maximum density of a subset of the integer grid avoiding the relevant $k$-in-a-row configurations, and let $3\mathbb{N}$ denote the positive multiples of $3$. Strict grid-density conjecture. For every $k\in 3\mathbb{N}$,

$$D(k,\mathbb{Z}^2)<1-\frac{2}{k}.$$

The source establishes the corresponding upper bound $D(k,\mathbb{Z}^2)\leq 1-2/k$ and reports strictness for $k=3$ and computational evidence for $k=6$; the conjecture remains open in general.