Weakley's domination-number equality conjecture for orders congruent to 1 modulo 4

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, and let $\gamma(\mathcal{Q}(n))$ denote its domination number. For $k \in \mathbb{N} \cup \{0\}$, consider the order $n=4k+1$. Weakley's conjecture. For all $k \in \mathbb{N} \cup \{0\}$,

$$2k+1=\gamma(\mathcal{Q}(4k+1)).$$

The conjecture asserts that the known lower bound is sharp for every nonnegative integer $k$; it is verified for $k \in \{1,\dots,32\}$, while the general case remains open.