Vilfred's non-distance-magic conjecture for cylindrical grid graphs

Let $P_m$ be the path graph on $m$ vertices and $C_n$ the cycle graph on $n$ vertices. A graph is Non-Distance Magic (NDM) if it admits no bijective labeling $f:V(G)\to\{1,2,\ldots,|V(G)|\}$ for which $\sum_{v\in N(u)}f(v)$ is constant over all vertices $u\in V(G)$. Vilfred's conjecture. For $m\geq 2$, $n\geq 3$ and $m,n\in\mathbb{N}$, the cylindrical grid graph $P_m\Box C_n$ is NDM. The paper proves this for several cases, including even $m$, but the general assertion remains open.