Existence conjecture for perfect distance-dominating sets with path components

Let $\Lambda_n$ be the $n$-dimensional lattice graph, and let $P_k$ denote the path on $k$ vertices. For a finite graph $H$, a $t$-PDDS$[H]$ in $\Lambda_n$ is a perfect distance-dominating set whose components are all isomorphic to $H$. The Cartesian product of graphs is denoted by $\square$.

Existence conjecture. Let $H$ be a finite path or a Cartesian product of two finite paths. Then a $t$-PDDS$[H]$ in $\Lambda_n$ exists if and only if at least one of the following holds: (i) $t=1$, $n\geq 2$, and $H=P_k$ for some $k\geq 1$; (ii) $t\geq 1$, $n=2$, and $H=P_k$ for some $k\geq 1$; (iii) $t\geq 1$, $n=2$, and $H=P_2\square P_k$ for some $k\geq 2$; (iv) $t=1$, $n=3r+2$ for some $r\geq 0$, and $H=P_2\square P_2$; or (v) $t=2$, $n=3$, and $H=P_2$.

This conjecture seeks to characterize the cases in which perfect distance-dominating sets with components that are paths or products of two paths exist. The source states that the authors can characterize such graphs only for $\Lambda_2$ and do not yet have enough evidence for the general case; the listed cases are proposed because they are strongly believed when $H$ has the specified form.