Type I conjecture for nontrivially unstable circulant graphs of order divisible by eight

Let $n$ be a positive integer divisible by $8$, and let $\Gamma=\operatorname{Cay}(\mathbb{Z}_{n},S)$ be a circulant graph with connection set $S$. Assume that $\Gamma$ is nontrivially unstable, meaning unstable in the paper's nontrivial sense, and recall that Type I is one of the paper's instability types.

Type I conjecture. Every such graph is of Type I.

The conjecture would rule out Type II examples among nontrivially unstable circulant graphs whose order is divisible by $8$. The supplied text does not provide evidence that this assertion has been proved or disproved.