Gross–Robbins–Tucker log-concavity conjecture for genus distributions

Let $G$ be a graph, and let $g_k(G)$ denote the number of equivalence classes of 2-cell embeddings of $G$ on the orientable surface of genus $k$. The sequence $\{g_k(G)\}_{k\geq 0}$ is the genus distribution of $G$. Gross–Robbins–Tucker conjecture. The genus distribution of every graph is log-concave. This conjecture proposes a general structural property for genus distributions and motivates the systematic study of their behavior across graph embeddings.