Perfect distance-dominating set conjecture for integer grids

Let $\Lambda_n$ be the integer grid with vertex set $Z^n$, with adjacency given by Lee distance $1$. Let $P_k$ be the path on $k$ vertices, and let an $r$-PDDS$[P_k]$ mean an $r$-perfect distance-dominating set whose components are isomorphic to $P_k$.

Perfect distance-dominating set conjecture. For $k\geq 1$, there is no $r$-PDDS$[P_k]$ in $\Lambda_n$ for $n\geq 3$ and $r>1$, with the exception of a $2$-PDDS$[P_2]$ in $\Lambda_3$.

This conjecture simultaneously extends the Golomb–Welch and diameter-perfect Lee-code questions. The stated exception corresponds to the known $DPL(3,6)$ construction, while the remaining nonexistence claims are open.