The edge-width four conjecture for 5-choosability on the torus

Let $G$ be a graph drawn on the torus. Its edge-width is the length of its shortest non-contractible cycle.

The edge-width four conjecture. Every graph drawn on the torus with edge-width at least four is $5$-choosable.

This would establish the main 5-choosability conjecture for an important class of toroidal graphs without short non-contractible cycles; it remains open in the source.