Classification conjecture for 5-regular Ricci-flat graphs

Let $G=(V,E)$ be a $5$-regular Ricci-flat graph. A graph is of Cartesian product type if it is the Cartesian product of a Ricci-flat $3$-regular graph and a Ricci-flat $2$-regular graph. Classification conjecture. If $G$ is a $5$-regular Ricci-flat graph, then $G$ is either isomorphic to $RF_{72}^{5}$ or is of Cartesian product type. The conjecture seeks a complete classification of $5$-regular Ricci-flat graphs; the existence of regular bone-idle graphs with odd vertex degree remains an open question.