  A graph is said to be walk-regular if, for each $\ell \geq 1$, every vertex
is contained in the same number of closed walks of length $\ell$. We construct
a $24$-vertex graph $H_4$ that is not walk-regular yet has maximized walk
entropy, $S^V(H_4,\beta) = \log 24$, for some $\beta>0$. This graph is a
counterexample to a conjecture of Benzi [Linear Algebra Appl.~443 (2014),
395--399, Conjecture 3.1]. We also show that there exist infinitely many
temperatures $\beta_0>0$ so that $S^V(G,\beta_0)=\log n_G$ if and only if a
graph $G$ is walk-regular.
