Rational-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. Rational-temperature conjecture. A graph $G$ is walk-regular if and only if there exists a rational $\beta>0$ such that

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

The preceding result shows that a non-walk-regular graph can attain maximum walk entropy at only finitely many positive temperatures, while the source offers this rational-temperature strengthening as a further conjecture.