Stahl's real-rootedness conjecture for graph genus polynomials

Let $G$ be a graph, and let

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

be its genus polynomial, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$.

Stahl's real-rootedness conjecture. For every graph $G$, the genus polynomial $\Gamma_G(x)$ has only real roots.

The conjecture was proposed by Stahl as a stronger property than log-concavity of the genus distribution. It was later disproved; the paper discusses counterexamples among graph genus polynomials, although it also notes an error in one previously proposed counterexample.