KMS quantum symmetric states

Let $A$ be a unital C$^*$-algebra and let $\sigma$ be a one-parameter automorphism group of $A$. We consider $\operatorname{QSS}_\sigma(A)$, the set of all quantum symmetric states on $*_1^\infty A$ that are also KMS states (for a fixed inverse temperature, for specificity taken to be $-1$) for the free product automorphism group $*_1^\infty\sigma$. We characterize the elements of $\operatorname{QSS}_\sigma(A)$, we show that $\operatorname{QSS}_\sigma(A)$ is a Choquet simplex whenever it is nonempty and we characterize its extreme points.