Hamiltonicity threshold conjecture for randomly perturbed directed graphs

Let $D$ be a directed graph on $n$ vertices with minimum semidegree $d$, and let $D(n,p)$ be the binomial random digraph obtained by adding each possible directed edge independently with probability $p$. Let $d=n/2-\eta$, where $1/2\leq\eta=\eta(n)=o(n)$.

Directed Hamiltonicity threshold conjecture. The $d$-threshold for Hamiltonicity in randomly perturbed directed graphs is $\eta/n^2$.

This extends the paper's threshold questions from graphs to directed graphs in the critical regime below semidegree $n/2$. The extension of the corresponding graph theorem to digraphs is stated to remain open.