The strict torus-to-grid density inequality for monochromatic k-in-a-row

Let $D(k,\mathbb{Z}^2)$ and $D(k, (\mathbb{Z}/k\mathbb{Z})^2)$ denote the corresponding maximum avoiding densities on the integer grid and the discrete torus, respectively. Torus-density conjecture. For every $k\in 3\mathbb{N}$,

$$D(k,(\mathbb{Z}/k\mathbb{Z})^2)<D(k,\mathbb{Z}^2).$$

The source records the general inequality in the opposite direction, gives strictness at $k=3$, and reports computational evidence at $k=6$; the proposed strict inequality remains open.