Conjecture on the limiting eigenvalue measure for randomly twisted transfer operators

Let $d\geq 2$, and let ${\mathscr L}_N$ be the randomly twisted transfer operator defined using independently uniform permutation matrices. Write $\mathrm{Sp}({\mathscr L}_N)$ for its spectrum, and let $C_0^\infty(\mathbb{C}^*)$ denote the smooth compactly supported test functions on $\mathbb{C}^*$. Limiting eigenvalue measure conjecture. There exists an absolutely continuous measure $\nu$, compactly supported in $\mathbb{C}$ and locally finite in $\mathbb{C}^*$, such that for every $\varphi\in C_0^\infty(\mathbb{C}^*)$,

$$\lim_{N\rightarrow\infty}\frac{1}{N}{\mathbb E}_N\left(\sum_{\lambda\in\mathrm{Sp}({\mathscr L}_N)}\varphi(\lambda)\right)=\int_{\mathbb{C}}\varphi\,d\nu.$$

This conjectures convergence of the expected normalized spectral counting measures, away from the origin, to an absolutely continuous limiting measure. The supplied text presents it as a natural and difficult problem following a deterministic eigenvalue-counting bound; no resolution is stated.