Type II classification conjecture for circulant graphs of order twice an odd integer

Let $m$ be an odd integer, and let $\Gamma=\operatorname{Cay}(\mathbb{Z}_{2m},S)$ be a circulant graph, where $S$ is its connection set. A graph is nontrivially unstable if it is unstable and its instability is not explained by a trivial automorphism; it is of Type II as in the paper's classification.

Type II classification conjecture. If $\Gamma$ is nontrivially unstable and of Type II, then

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

The conjecture would identify all nontrivially unstable Type II circulant graphs when the order is twice an odd integer. The paper cites results for odd prime-power parameters and leaves the general odd case as the proposed classification.