Strong log-convexity of genus sequences

For a graph G, and a nonnegative integer g, let ag (G) be the number of 2-cell embeddings of G in an orientable surface of genus g (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker Genus distributions for bouquets of circles, J. Combin. Theory Ser. B 47 (1989), 292-306 proposed a conjecture that the sequence a₀ (G), a₁ (G), a₂ (G), is log-concave for every graph G. This conjecture is reminiscent to the Heron-Rota-Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al. , except that it is closer to the notion of -matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.