Linear oriented Ramsey number of sparse random digraphs

Fix $d\geq 2$, and let $H=\ovar{G}(n,d)$ be the sparse random digraph used in the paper. Write $\ovar{r_1}(H)$ for its one-color oriented Ramsey number. Linear random-digraph conjecture. With high probability,

$$\ovar{r_1}(H)=O_d(n).$$

The paper currently proves only an upper bound of $n(\log n)^{O_d(1)}$ for fixed $d$ and leaves the asserted linear bound open.