Beaton–Brown average dominating-set order conjecture

Let $G$ be a graph with $n\geq 2$ vertices, and let $\operatorname{avd}(G)$ denote its average order of dominating sets. If $G$ has no isolated vertices, then Beaton–Brown's conjecture.

$$\operatorname{avd}(G)\leq \frac{2n}{3}.$$

The conjecture sharpens the previously known upper bound $3n/4$ for graphs without isolated vertices. The supplied status evidence indicates that the result is proved in this paper.