Unimodality conjecture for the choice–chromatic gap

Let $\jump(g,r)$ be the maximum choice–chromatic gap among graphs embeddable on the orientable surface of genus $g$ with chromatic number $r$, and let $H(g)$ be the maximum chromatic number of a graph embeddable on that surface. Unimodality conjecture. For each fixed $g$, the function $\jump(g,r)$ is unimodal in $r$: there exists $r_0=r_0(g)$ such that

$$\jump(g,1)\leq\jump(g,2)\leq\dotsb\leq\jump(g,r_0)\geq\dotsb\geq\jump(g,H(g)).$$

The conjecture is known for planar and toroidal graphs, and the source notes that unimodality also holds for the projective plane and Klein bottle; the general orientable-surface case remains open.