Classification conjecture for regular graphs of complexity less than one

Let $G$ be a connected regular graph, and let $c(G)$ denote its complexity. Let $C_7$ be the cycle of length seven, and let $K_{n,n}$ and $K_{2n}$ denote the complete bipartite and complete graphs, respectively.

Regular-graph complexity conjecture. If $c(G)<1$, then $G$ is either isomorphic to $C_7$, or there is an integer $n\geq2$ such that $G$ is isomorphic to either $K_{n,n}$ or $K_{2n}$.

The conjecture proposes that, apart from these specified cases, connected regular graphs have complexity at least one; the source gives no resolution status.