Wilson's conjecture on non-trivially unstable Rose Window graphs

Let $n$ be an integer with $n\geq 3$, and let $a,r\in\mathbb{Z}_n$ with $a,r\ne 0$. The Rose Window graph $R_n(a,r)$ has vertex set $\{u_i,v_i:i\in\mathbb{Z}_n\}$ and edges

$$\{u_i,u_{i+1}\},\quad \{v_i,v_{i+r}\},\quad \{u_i,v_i\},\quad \{u_{i+a},v_i\},\qquad i\in\mathbb{Z}_n,$$

where subscripts are computed modulo $n$. A graph is non-trivially unstable if it is unstable with respect to its canonical double cover and is neither bipartite nor has two vertices with the same neighborhood. The nine families W1--W9 are the families of unstable Rose Window graphs listed in Table 1 of the source.

Wilson's conjecture. Every non-trivially unstable Rose Window graph is isomorphic to a graph in one of the families W1--W9.

Wilson checked this conjecture computationally for Rose Window graphs on at most $200$ vertices. The conjecture asserts that the nine listed families exhaust all non-trivially unstable Rose Window graphs; the source gives no resolution beyond this computational verification.