The Log-Concavity Genus Distribution conjecture for graph genus polynomials

Let $G$ be a graph. Its genus polynomial is

$$\Gamma_G(x)=\sum_{k\geq 0}g_k(G)x^k,$$

where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$. The polynomial is log-concave when its coefficient sequence satisfies

$$(g_k(G))^2\geq g_{k-1}(G)g_{k+1}(G)$$

for every $k\in\mathbb{N}$.

Log-Concavity Genus Distribution conjecture. For every graph $G$, the genus polynomial is log-concave.

The conjecture was proposed in 1989 and has been confirmed for various classes of graphs and shown to be preserved under several graph amalgamation operations. It was nevertheless later disproved by a counterexample, so its database status is refuted.