The toroidal efficient total coloring conjecture

Let $M(G)$ be a toroidal cubic map whose 1-skeleton is a simple cubic graph of girth $4$. Call $G$ toroidally 3-edge connected when the bicutout condition defined in the source holds, and call a belt a cycle of the type considered in the source. Toroidal efficient total coloring conjecture. If $M(G)$ is toroidally 3-edge connected and every belt has length divisible by $4$, then $G$ admits an efficient total coloring compatible with $M(G)$. The supplied text gives no resolution or supporting context beyond this assertion.