Unimodality conjecture for dominating-set sequences of finite graphs

Let $G$ be a finite undirected graph on $n$ vertices. A dominating set is a set of vertices such that every vertex is either in the set or adjacent to a vertex in the set. For $0\leq j\leq n$, let $d_j(G)$ be the number of dominating sets of $G$ with cardinality $j$. Unimodality conjecture. The sequence $d_0(G),\ldots,d_n(G)$ is unimodal: there exists some $0\leq m\leq n$ such that

$$d_0(G)\leq d_1(G)\leq\cdots\leq d_m(G)\geq d_{m+1}(G)\geq\cdots\geq d_n(G).$$

The conjecture concerns the shape of dominating-set enumerators, while this paper gives examples showing that dominating-set sequences need not be log-concave. Since log-concavity implies unimodality under the relevant endpoint conditions, disproving log-concavity does not by itself disprove this conjecture; the supplied text gives no resolution status for the unimodality claim.