Canonical double covers of circulants

The canonical double cover $B(X)$ of a graph $X$ is the direct product and $K_2$. If $Aut(B(X)) \cong Aut(X) \times \mathbb{Z}_2$ then called stable; otherwise unstable. An unstable nontrivially if it connected, non-bipartite distinct vertices have different neighborhoods. Circulant Cayley on cyclic group. Qin et al. conjectured in [J. Combin. Theory Ser. B 136 (2019), 154-169] that there are no nontrivialy circulants odd order. In this paper we prove conjecture.