The local \overline{d}-continuity conjecture for mixing matrix equilibrium states

Let $M_d(\mathbb{R})^M$ denote the space of $M$-tuples of real $d\times d$ matrices. For each irreducible $\mathsf{B}\in M_d(\mathbb{R})^M$ and $t>0$, let $\mu_{\mathsf{B},t}$ denote the unique equilibrium state of $(\mathsf{B},t)$. Suppose that $\mathsf{A}\in M_d(\mathbb{R})^M$ is irreducible and that $\mu_{\mathsf{A},s}$ is mixing.

Local \overline{d}-continuity conjecture. The map

$$(\mathsf{B},t)\longmapsto \mu_{\mathsf{B},t}$$

is $\overline{d}$-continuous at $(\mathsf{A},s)$.

This strengthens parameter continuity at points whose equilibrium state is mixing, allowing both the matrix tuple and the parameter to vary. The paper states it as a conjecture and supplies no resolution.