Unstable circulants of twice an odd order

Let $m>1$ be an odd integer, let $n=2m$, and let $S\subseteq\mathbb{Z}_{2m}$ define the connected, non-bipartite circulant graph

$$\Gamma=\operatorname{Cay}(\mathbb{Z}_{2m},S).$$

The conjecture. The graph $\Gamma$ is unstable if and only if either there exists a nonzero $h\in 2\mathbb{Z}_{2m}$ such that

$$S\cap(2\mathbb{Z}_{2m}+h)=S\cap 2\mathbb{Z}_{2m},$$

or

$$\operatorname{Cay}(\mathbb{Z}_{2m},S)\cong\operatorname{Cay}(\mathbb{Z}_{2m},S+m).$$

This is presented as a suggested extension of the paper's square-free-order result to arbitrary odd $m>1$. The supplied text gives no evidence that the assertion has been proved or disproved.