Grünbaum's toroidal conjecture on non-3-edge-colorable cubic graphs

Let $G$ be a $2$-connected cubic graph embedded in the torus. Grünbaum's conjecture. If $G$ is not $3$-edge-colorable, then $G$ contains two edges on the same face of the embedding whose removal gives a planar graph in which the four endpoints of the removed edges lie on the boundaries of two faces. The conjecture is false in general for surfaces of genus $5$, as shown by Kochol, but the paper proves the toroidal case.