  Let $\mathscr{H}$ be a family of digraphs. A digraph $D$ is
\emph{$\mathscr{H}$-free} if it contains no isomorphic copy of any member of
$\mathscr{H}$. For $k\geq2$, we set $C_{\leq k}=\{C_{2}, C_{3},\ldots,C_{k}\}$,
where $C_{\ell}$ is a directed cycle of length $\ell\in\{2,3,\ldots,k\}$. Let
$D_{n}^{k}(\xi,\zeta)$ denote the family of \emph{${C}_{\le k}$-free} strong
digraphs on $n$ vertices with every vertex having out-degree at least $\xi$ and
in-degree at least $\zeta$, where both $\xi$ and $\zeta$ are positive integers.
Let $\varphi_{n}^{k}(\xi,\zeta)=\max\{|A(D)|:\;D\in D_{n}^{k}(\xi,\zeta)\}$ and
$\Phi_{n}^{k}(\xi,\zeta)=\{D\in D_{n}^{k}(\xi,\zeta):
|A(D)|=\varphi_{n}^{k}(\xi,\zeta)\}$. Bermond et al.\;(1980) verified that
$\varphi_{n}^{k}(1,1)=\binom{n-k+2}{2}+k-2$. Chen and Chang\;(2021) showed that
$\binom{n-1}{2}-2\leq\varphi_{n}^{3}(2,1)\leq\binom{n-1}{2}$. This upper bound
was further improved to $\binom{n-1}{2}-1$ by Chen and Chang\;(DAM, 2022),
furthermore, they also gave the exact values of $\varphi_{n}^{3}(2,1)$ for
$n\in \{7,8,9\}$. In this paper, we continue to determine the exact values of
$\varphi_{n}^{3}(2,1)$ for $n\ge 10$, i.e.,
$\varphi_{n}^{3}(2,1)=\binom{n-1}{2}-2$ for $n\geq10$.
