Bordenave–Chafaï conjecture for random regular digraph matrices

Let $A_n$ be a random regular digraph matrix, meaning a uniformly random element of $\mathop{\mathrm{M}}_n(d)$, where $\mathop{\mathrm{M}}_n(d)$ is the set of $n\times n$ zero-one matrices whose row and column sums are all $d$. For a matrix $M$, let $\mu_M$ denote its empirical spectral distribution. Bordenave–Chafaï conjecture. (1) If $d=d_n\le n/2$ and $d\rightarrow\infty$ as $n\rightarrow\infty$, then $\mu_{\frac{1}{\sqrt{d}}A_n}$ converges to the circular law. (2) If $d\ge 3$ is fixed independently of $n$, then $\mu_{A_n}$ converges to the oriented Kesten–McKay law on $\mathbb{C}$, with density

$$f_{KM}(w)=\frac{1}{\pi}\frac{d^2(d-1)}{(d^2-|w|^2)^2}\mathbf{1}_{\\{|w|\le \sqrt{d}\\}}.$$

These claims predict the limiting spectral distributions of random regular digraph adjacency matrices in both the growing-degree and fixed-degree regimes. The statement is presented as an augmented version of a conjecture of Bordenave and Chafaï; the source does not provide evidence of resolution.