The monotonicity sandwich conjecture for average dominating-set order

Let $G$ be a nonempty graph. For a vertex $v$ and an edge $e$, write $G-v$ and $G-e$ for the graphs obtained by deleting $v$ and $e$, respectively. The monotonicity sandwich conjecture. There exist a vertex $v$ and an edge $e$ such that

$$\operatorname{avd}(G-v)<\operatorname{avd}(G)<\operatorname{avd}(G-e).$$

Although deleting a vertex or edge always decreases the number of dominating sets, it need not make the average order of a dominating set monotone in either direction. The conjecture is reported to hold for all graphs on at most $7$ vertices; its validity for all nonempty graphs remains open.