The distinguishing-index conjecture for connected finite regular graphs

Let $G$ be a connected, finite, regular graph, and let $D'(G)$ denote the least number of colours needed to colour the edges of $G$ so that the only colour-preserving automorphism is the identity. The regular-graph distinguishing-index conjecture. Then $D'(G)\leq 2$, unless $G$ is either $K_n$ for $n\leq 5$, $K_{n,n}$ for $n\leq 3$, or $C_5$. This would improve the general bound for distinguishing indices in the regular case; the paper proves the weaker general estimate but presents this sharper assertion as conjectural.