Odd-order conjecture for nontrivially unstable circulants

A circulant is a Cayley graph of a cyclic group; write an order-$n$ circulant as $\operatorname{Cay}(\mathbb{Z}_n,S)$. A circulant is nontrivially unstable when it is unstable but does not arise from the trivial sources of instability considered in the paper. Odd-order conjecture. There is no nontrivially unstable circulant of odd order. This is a weaker conjecture than Wilson's proposed classification and is presented as a first step toward classifying all nontrivially unstable circulants. The paper does not state a resolution of this conjecture.