Qin–Xu–Zhou's conjecture on nontrivially unstable circulants of odd order

Let $X$ be a circulant, that is, a Cayley graph on a cyclic group. Its canonical double cover $B(X)$ is the bipartite graph with vertex set $V(X)\times\mathbb{Z}_2$ in which $(x,0)$ is adjacent to $(y,1)$ exactly when $x$ and $y$ are adjacent in $X$. The graph $X$ is unstable if $B(X)$ has automorphisms that do not preserve the two canonical bipartition classes, and nontrivially unstable if it is unstable without belonging to the trivial exceptional class described in the source. Qin–Xu–Zhou's conjecture. There is no nontrivially unstable circulant of odd order. This conjecture concerns the classification of vertex-transitive unstable graphs. It is known that every circulant of odd prime order is stable, but the assertion for all odd orders remains open.