Estrada's walk-regularity conjecture at temperature one

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. Estrada's conjecture. A graph $G$ is walk-regular if and only if

$$S^V(G,1)=\log n_G.$$

The paper proves that some positive temperature has this characterization, but does not identify such a temperature explicitly; temperature $1$ is presented as an open conjectural choice.