  Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum
in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For
$k:=\min\{k_1,k_2\}\ge 2$ and $M/N>\max\{k_1,k_2\}$, $M=\Theta(N)$, we show
that, among those digraphs, the fraction of $k$-strongly connected digraphs is
$1-O\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp
edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in
the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$
with probability $1-O\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core exists and
is $k$-strongly connected.
