The toroidal choice–chromatic gap conjecture for 3-chromatic graphs

Let $\jump(g,r)$ denote the maximum of the difference between list-chromatic number and chromatic number over graphs embeddable on the orientable surface of genus $g$ with chromatic number $r$. A graph is toroidal when it is embeddable on the orientable surface of genus $1$. Toroidal gap conjecture.

$$\jump(1,3)=2.$$

This is equivalent to asserting that every $3$-chromatic toroidal graph is $5$-choosable. Resolving it would determine the maximum choice–chromatic gap for toroidal graphs; the source presents it as open.