  Consider the matrix $A_{\mathcal{G}}$ chosen uniformly at random from the
finite set of all $N$-dimensional matrices of zero main-diagonal and binary
entries, having each row and column of $A_{\mathcal{G}}$ sum to $d$. That is,
the adjacency matrix for the uniformly random $d$-regular simple digraph
$\mathcal{G}$. Fixing $d \ge 3$, it has long been conjectured that as $N \to
\infty$ the corresponding empirical eigenvalue distributions converge weakly,
in probability, to an explicit non-random limit, %measure $\mu_d$ on
$\mathbb{C}$, which is given by the Brown measure of the free sum of $d$ Haar
unitary operators. We reduce this conjecture to bounding the decay in $N$ of
the probability that the minimal singular value of the shifted matrix $A(w) =
A_{\mathcal{G}} - w I$ is very small. While the latter remains a challenging
task, the required bound is comparable to the recently established control on
the singularity of $A_{\mathcal{G}}$. The reduction is achieved here by sharp
estimates on the behavior at large $N$, near the real line, of the Green's
function (aka resolvent) of the Hermitization of $A(w)$, which is of
independent interest.
