The mixing conjecture for matrix equilibrium states

Let $M_d(\mathbb{R})^M$ denote the space of $M$-tuples of real $d\times d$ matrices, and let $\mathsf{A}\in M_d(\mathbb{R})^M$ be irreducible. For $s>0$, write the equilibrium state of $(\mathsf{A},s)$ for the corresponding equilibrium state on the symbolic space, with shift map $\sigma$.

Mixing conjecture. If for some $s>0$ the equilibrium state of $(\mathsf{A},s)$ is ergodic with respect to $\sigma^d$, then it is mixing with respect to $\sigma$. If there exists $s>0$ such that the equilibrium state of $(\mathsf{A},s)$ is mixing with respect to $\sigma$, then for every $s>0$ the equilibrium state of $(\mathsf{A},s)$ is mixing with respect to $\sigma$.

The conjecture seeks a complete characterization of mixing for matrix equilibrium states, a problem related to when products of an irreducible matrix tuple become reducible. The authors note that the question appears accessible in dimension two but is less clear in higher dimensions.