Universality conjecture for diagonal resolvent entries of rotationally invariant random matrices

Let $\vb{M}$ be a rotationally invariant random matrix of size $N$ with a non-pathological limiting spectral density. For $z$ outside the spectrum of $\vb{M}$, define

$$[\vb{G}_{\vb{M}}^N(z)]_{11}:=[(z\vb{1}-\vb{M})^{-1}]_{11}.$$

Let $\mathfrak{g}_{\vb{M}}(z)$ denote the limiting Stieltjes transform and let $\widetilde{\mathcal{O}}(z)$ denote the rescaled self-overlap corresponding to $z$. Universality conjecture. As $N\to\infty$, the distribution of $[\vb{G}_{\vb{M}}^N(z)]_{11}$ converges in law to a complex Student law with $\nu=2$ and density

$$\omega\mapsto\frac{1}{\pi}\frac{\beta}{(\beta+|\omega-\mathfrak{g}_{\vb{M}}(z)|)^2},\qquad \beta=\pi\widetilde{\mathcal{O}}(z).$$

Equivalently, the distribution of

$$\Omega:=\frac{[\vb{G}_{\vb{M}}^N(z)]_{11}-\mathfrak{g}_{\vb{M}}(z)}{\sqrt{\pi\widetilde{O}(z)}}$$

should be independent of $\vb{M}$ and $z$, with density $\omega\mapsto \frac{1}{\pi(1+|\omega|^2)^2}$. This conjecture proposes a universal limiting law for diagonal resolvent entries beyond the Ginibre case; its resolution is not established by the supplied text.