Log-concavity conjecture for genus distributions of surface-triangulating graphs

Let $G$ be a graph that triangulates some surface, and let its genus distribution be the sequence of numbers of $2$-cell embeddings of $G$ in orientable surfaces of each genus. Surface-triangulating graph conjecture. Every graph that triangulates some surface has log-concave genus distribution. The source presents this as another plausible conjecture following its discussion of genus distributions and counterexamples to the general log-concavity conjecture; no resolution is given.