Grünbaum's torus conjecture for cubic graph colorings

Let $G$ be a $2$-connected cubic graph embedded in the torus. The graph $G$ is 3-edge-colorable if its edges can be colored with three colors so that adjacent edges receive different colors. Grünbaum's torus conjecture. If $G$ is not 3-edge-colorable, then $G$ contains two edges whose removal gives a planar graph. Grünbaum's broader conjecture for cubic graphs with polyhedral embeddings in orientable surfaces was disproved in genus $5$ by Kochol. The torus case remains unresolved; the conjecture is motivated by examples of toroidal snarks obtained from Petersen-graph constructions.