Counting strongly connected $(k_1,k_2)$-directed cores

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 $M\mathrm{/}N>\max \{k_1,k_2\}$, $M=\mathrm{\Theta }(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^{\ast }(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^{\ast }(k_1,k_2)$ with probability $1-O(N^{-(k-1)})$ the giant $(k_1,k_2)$-core exists and is $k$-strongly connected.