Unimodality conjecture for graph genus distributions

For a graph $G$ and a nonnegative integer $g$, let $a_g(G)$ denote the number of $2$-cell embeddings of $G$ in an orientable surface of genus $g$, counted up to combinatorial homeomorphism equivalence. The sequence $(a_0(G),a_1(G),a_2(G),\dots)$ is the genus distribution of $G$. Unimodality conjecture. The genus distribution of every graph $G$ is unimodal. The paper notes that although the log-concavity conjecture has counterexamples, the genus distributions in those examples remain unimodal; the conjecture is presented as a plausible weakening and remains open.