Universal limiting law conjecture for the normalized Stieltjes-transform fluctuations

Let $\vb{M}$ be a rotationally invariant random matrix of size $N$ with a well-behaved limiting spectral density. For $z$ inside the spectrum, let $\mathfrak{g}_{\vb{M}}$ be the limiting Stieltjes transform and let $\rho_{\vb{M}}$ be the limiting spectral distribution. Define

$$Z:=\lim_{N\to+\infty}\sqrt{N}\,\frac{\mathfrak{g}_{\vb{M}}^N(z)-\mathfrak{g}_{\vb{M}}(z)}{\sqrt{\pi\rho_{\vb{M}}(z)}}.$$

Universal fluctuation conjecture. The distribution of $Z$ is independent of $\vb{M}$ and $z$, satisfies

$$P(|Z|\geq r)\underset{r\to+\infty}{\sim}\frac{1}{r^2},$$

and is invariant under the transformation $Z\mapsto\sqrt{2}/Z$. The conjecture predicts a universal heavy-tailed law for normalized Stieltjes-transform fluctuations, but the supplied text does not establish its resolution.