Wilson's classification conjecture for nontrivially unstable circulants

Let $G$ be a group and let $S$ be a nonempty inverse-closed subset of $G\setminus\{1\}$, where $1$ is the identity. The Cayley graph $\operatorname{Cay}(G,S)$ has vertex set $G$, with $x$ and $y$ adjacent exactly when $yx^{-1}\in S$. A circulant is a Cayley graph of a cyclic group; write it as $\operatorname{Cay}(\mathbb{Z}_n,S)$. Such a circulant is nontrivially unstable when it is unstable but does not fall under the trivial sources of instability described in the paper. Conditions (C.1)--(C.4) are the four conditions listed in the source: (C.1) $n$ is even and has an even divisor $a$ such that every even $s\in S$ also satisfies $s+a\in S$; (C.2) $n$ is divisible by $4$ and has an odd divisor $b$ such that every odd $s\in S$ also satisfies $s+2b\in S$; (C.3) $n$ is even and there is a subgroup $H\leq\mathbb{Z}_n$ for which $R:=\{j\bmod n\mid j\in S,\ j+H\not\subseteq S\}\neq\emptyset$, $D:=\gcd(R)>1$, and $j/D$ is odd for every $j\in R$; (C.4) $n$ is even and some integer $g$ coprime to $n$ satisfies $gs+n/2\in S$ for every $s\in S$. Wilson's classification conjecture. Every nontrivially unstable circulant $\operatorname{Cay}(\mathbb{Z}_n,S)$ satisfies at least one of (C.1)--(C.4). The conjecture proposes a classification of all nontrivially unstable circulants, but the paper notes that the proposed classification is false after correcting (C.2), and no conjectural classification is ultimately known.