Randomised twisted Toeplitz spectral-measure conjecture

Let $a(x,z)$ be the symbol on $[0,1]\times\mathbb{T}$

$$a(x,z)=\sum_{j=-q}^p a_j(x)z^j,$$

where $a_j(x)$ are continuous complex-valued functions on $[0,1]$. Let $\mathcal{T}_n(a)$ be the randomised twisted Toeplitz matrix defined in the source, and let $\nu_x$ be the weak limit of the eigenvalue-counting measures of the frozen Toeplitz matrices associated with $a_x=a(x,\cdot)$. Define

$$\mu_n=\frac{1}{n}\sum_{\lambda\in\operatorname{sp}(\mathcal{T}_n(a))}\delta_\lambda.$$

Randomised twisted Toeplitz spectral-measure conjecture. The measures $\mu_n$ converge weakly to

$$\mu=\int_0^1\nu_x\,dx.$$

This conjecture predicts the same averaged frozen-symbol limit for random sampling points in the symbol as for structured banded perturbations. The supplied source does not state whether the conjecture has been resolved.