Unimodality of the genus distribution of graphs

Prove that the genus distribution of every graph is unimodal, meaning that its sequence of genus counts first weakly increases and then weakly decreases.

Background

The paper discusses the shape of genus distributions for ordinary graphs as context for its conjectures about signed graphs. The log-concavity conjecture for graph genus distributions has been disproved, but the cited counterexamples remain unimodal.

The authors explicitly report that the unimodality conjecture for graph genus distributions remains unresolved; this is presented as a related open problem rather than as a result established in the paper.