The King grid exponential domination lower-bound conjecture

Let $\mathcal{K}_n=P_n\boxtimes P_n$ be the King grid, where $P_n$ is the path on $n$ vertices. Write $\gamma^*_e(G)$ for the minimum cardinality of an exponential dominating set in a graph $G$. The King grid exponential domination conjecture. For all $n$,

$$\left\lceil \frac{n^2}{23} \right\rceil \le \gamma^*_e(\mathcal{K}_n).$$

The conjecture would match the construction establishing asymptotic density at most $1/23$ and the resulting upper bound for sufficiently large King grids. No matching lower bound or resolution is given in the supplied source.