Two-label conjecture for distinguishing indices of regular graphs

Let $G$ be a $k$-regular graph, meaning that every vertex of $G$ has degree $k$. The distinguishing index of $G$, denoted $D'(G)$, is the least number of edge labels needed so that only the identity automorphism preserves the labeling. Two-label conjecture. If $k\geq 5$, then

$$D'(G)\leq 2.$$

This conjecture proposes a uniform two-label bound for distinguishing regular graphs of degree at least five. The source gives no evidence of a resolution.