Canonical double covers of circulants

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