Oriented Kesten–McKay law conjecture for random regular digraphs

Fix an integer $d\ge 3$, and let $A^{(d)}_n$ be uniformly drawn from the set $\mathcal{A}_{n,d}$ of adjacency matrices of $d$-regular directed graphs on $n$ vertices. Let $\mu_{A^{(d)}_n}$ denote its empirical spectral distribution, and let $\mu^{(d)}_{\mathrm{KM}}$ be the oriented Kesten–McKay law on $\mathbb{C}$ with density

$$f^{(d)}_{\mathrm{KM}}(z)=\frac{1}{\pi}\frac{d^2(d-1)}{(d^2-|z|^2)^2}\mathbf{1}_{\{|z|\le\sqrt d\}}$$

with respect to Lebesgue measure. Oriented Kesten–McKay law conjecture. As $n\to\infty$, $\mu_{A^{(d)}_n}\to\mu^{(d)}_{\mathrm{KM}}$ in probability. For fixed degree, this predicts the limiting empirical spectral distribution of random regular digraph adjacency matrices; the source calls it well known but provides no resolution in the supplied text.