Šoltés graph uniqueness conjecture

Let $C_{11}$ denote the cycle on eleven vertices. A Šoltés graph is a graph for which deleting any vertex leaves the Wiener index unchanged. Šoltés graph uniqueness conjecture. The cycle on eleven vertices, $C_{11}$, is the only Šoltés graph. The source notes that this is reinforced by the absence of Šoltés graphs among vertex-transitive graphs with fewer than $48$ vertices, but gives no resolution.