Hussain–Niepel–Kinawi's optimal LPDS density conjecture for king grids

Let $G=(V,E)$) be the infinite king grid with vertex set $V=\mathbb{Z}\times\mathbb{Z}$, in which two vertices are adjacent exactly when their Euclidean distance is at most $\sqrt{2}$. For a set $S\subseteq V$, define its density by

$$D(S)=\limsup_{k\rightarrow\infty}\frac{|S\cap N_G^k[u]|}{|N_G^k[u]|},$$

where $N_G^k[u]=\{x\in V(G)\mid d(u,x)\leq k\}$ and $d(u,x)$ is graph distance. An optimal locating-paired-dominating set (optimal LPDS) is an LPDS of minimal density. Hussain–Niepel–Kinawi's conjecture. If $S$ is an optimal LPDS in the king grid and $D(S)$ is its density, then

$$D(S)=\frac{2}{9}.$$

Previously, the density of an optimal LPDS was known to satisfy $3/14\leq D(S)\leq 2/9$; the conjecture asserts that the upper bound is attained and hence determines the optimal density exactly.