Asymptotic cycle-count conjecture for k-geodetic digraphs

Let $ex(n;C_{k+1};k)$ denote the maximum number of directed copies of the cycle $C_{k+1}$ in a $k$-geodetic digraph of order $n$. The asymptotic cycle-count conjecture asserts that, for every $k\geq 2$,

$$ex(n;C_{k+1};k)=\frac{1}{k+1}n^{\frac{k+1}{k}}+O\left(n^{\frac{1}{k}}\right).$$

This extends the proved triangle case $ex(n;C_3;2)=\frac{1}{3}n^{3/2}+O(n^{1/2})$ and predicts that permutation digraphs are asymptotically extremal for counting directed $(k+1)$-cycles in $k$-geodetic digraphs; the general case remains open.