Bound on maximum-entropy temperatures for non-walk-regular graphs

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 $\beta>0$. Temperature-count conjecture. If $G$ is not walk-regular, then there are at most $n_G-1$ values $\beta>0$ such that

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

The paper proves only that this set of temperatures is finite. The conjecture proposes the sharper bound $n_G-1$ in terms of the number of vertices.