Banded perturbation conjecture for twisted Toeplitz matrices

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 $R_n(a)=T_n(a)+\sigma_nX_n$ be the structured random perturbation described above, and let $\nu_x$ be the weak limit of the eigenvalue-counting measures of the frozen Toeplitz matrices $T_n(a_x)$, where $a_x=a(x,\cdot)$. Define

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

Banded perturbation conjecture. The measures $\mu_n$ converge weakly to

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

This conjecture extends the tridiagonal random-perturbation spectral asymptotics to banded twisted Toeplitz matrices, predicting that the limiting eigenvalue distribution is obtained by averaging the limiting measures of the frozen symbols. Its status is not resolved in the supplied source.