Odd-height cylindrical grid graphs are non-distance-magic

Let $P_{2k+1}$ be the path graph on $2k+1$ 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 $u\in V(G)$. Odd-height grid conjecture. For $n\geq 3$ and $n,k\in\mathbb{N}$, the graph $P_{2k+1}\Box C_n$ is NDM. The preceding results establish the corresponding even-height case, leaving this odd-height case as the remaining conjecture; its resolution is not given in the supplied text.