The toroidal 5-choosability and 5-colorability conjecture

A graph drawn on the torus is a toroidal graph, and it is 5-choosable if every assignment of lists of at least five colors to its vertices admits a proper coloring from those lists. It is 5-colorable if it admits a proper coloring with five colors.

The toroidal 5-choosability conjecture. A toroidal graph is $5$-choosable if and only if it is $5$-colorable.

The paper establishes this equivalence under additional structural assumptions, while the unrestricted statement remains open.