Benzi's single-temperature walk-regularity conjecture

Let $G$ be a simple graph, let $n_G$ be its number of vertices, and let $S^V(G,\beta)$ denote its walk entropy at temperature $\beta\geq0$. The graph $G$ is walk-regular when every vertex is contained in the same number of closed walks of every length. Benzi's conjecture. A graph $G$ is walk-regular if and only if there exists $\beta>0$ such that

$$S^V(G,\beta)=\log n_G.$$

Benzi's earlier theorem proves the implication from equality of the walk entropy on any set of real temperatures having an accumulation point; this conjecture asks whether one positive temperature suffices and remains open in the source.