Vertex-transitivity conjecture for Šoltés graphs

Let $G$ be a graph, and let $W(G)$ denote its Wiener index. A Šoltés graph is a graph for which deleting any vertex leaves the Wiener index unchanged. A graph is vertex transitive if its automorphism group acts transitively on its vertices. Vertex-transitivity conjecture. If $G$ is a Šoltés graph, then $G$ is vertex transitive. The motivation is that vertex deletion can produce equal Wiener indices on vertices in the same orbit, but the source gives no resolution.