Weak $k$-metric dimension formula for Hamming graphs $K_n\square K_m$

Let $K_n\square K_m$ be the Cartesian product of complete graphs, with $n\ge 3$, $m\ge n+1$, and $3\le k\le 2n$. The weak $k$-metric dimension formula asserts that

$$\operatorname{wdim}_{k}(K_n\square K_m)= \begin{cases} m\left\lceil\frac{k}{2}\right\rceil,& \text{if }k\text{ is even},\\[0.2cm] m\left\lceil\frac{k}{2}\right\rceil-1,& \text{if }k\text{ is odd}. \end{cases}$$

This conjecture is motivated by computational results, an integer-linear-programming bound, and formulas established for related parameter ranges; the source notes that a proof may require lengthy considerations similar to those used for the corresponding theorem.