The circulant orbit–eigenvalue conjecture

Let $G$ be an uncoloured circulant graph. Write $V*V$ for the set of unordered pairs of vertices of $G$, and let $\operatorname{Aut}(G)$ act on $V*V$ through its action on vertices. Let $r$ be the number of orbits of this action and let $s$ be the number of distinct eigenvalues of the adjacency matrix of $G$.

Circulant orbit–eigenvalue conjecture. The number of orbits of the action of $\operatorname{Aut}(G)$ on $V*V$ coincides with the number of distinct eigenvalues of its adjacency matrix; equivalently, $r=s$.

This is stated as equivalent to the preceding conjecture in the uniform-coloured setting. The source gives no resolution, so the claim remains open here.