Matrix Stahl–Totik regularity conjecture for matrix-valued measures

Let $E$ be a finite union of disjoint closed intervals in ${\mathbb{R}}$. Suppose $\mu$ is an $l\times l$ matrix-valued measure on ${\mathbb{R}}$ with

$$\sigma_{\mathrm{ess}}(d\mu)=E.$$

For each $\eta>0$ and $m=1,2,\dots$, define

$$S_{m,\eta}=\left\{x\colon \mu\left([x-\tfrac{1}{m},x+\tfrac{1}{m}]\right)\geq e^{-\eta m}{\boldsymbol{1}}\right\}.$$

Suppose that for each $\eta$, with $|\cdot|$ denoting Lebesgue measure,

$$\lim_{m\to\infty}|E\setminus S_{m,\eta}|=0.$$

Matrix Stahl–Totik conjecture. Then $\mu$ is regular.

This conjecture is the matrix-valued analogue of a theorem of Stahl and Totik. It is known for direct sums by the cited scalar result, while the general matrix case is expected to require a matrix Remez inequality.