The torus exponential domination lower-bound conjecture

Let $C_m$ and $C_n$ be cycle graphs, and let $C_m\square C_n$ be their Cartesian product. Write $\gamma^*_e(G)$ for the minimum cardinality of an exponential dominating set in a graph $G$. The torus exponential domination conjecture. For all $m$ and $n$,

$$\left\lceil\frac{mn}{13}\right\rceil \le \gamma^*_e(C_m\square C_n).$$

Together with the known asymptotic construction giving density at most $1/13$, this would determine the optimal density for exponential domination of torus graphs. The stated lower bound is motivated by the existing counting lower bound with denominator $15.875$ and remains unresolved in the supplied source.