The free state for the Potts model on Cayley trees is either extremal or glassy

For the Potts model on the Cayley tree ${\mathbb{T}}^d$ with branching factor $d\ge 2$, we consider the free state which is obtained as the limiting Gibbs measure with free boundary conditions. We prove that the free state is either extremal or glassy (i.e., whose decomposition into extremal Gibbs measures contains uncountably many components). As a corollary, the free state for the Ising model on ${\mathbb{T}}^d$ is glassy if and only if the inverse temperature $\beta >\mathrm{arctanh}(1\mathrm{/}\sqrt{d})$; this generalizes a previous result by Gandolfo, Maes, Ruiz and Shlosman (2020) from a very low temperature regime to the entire spin-glass regime.