Stein's minimum semi-degree conjecture for oriented paths

Let $D$ be an oriented graph, let $k$ be a positive integer, and write $\delta^{0}(D)$ for the minimum semi-degree. An oriented path of length $k$ is a path on $k+1$ vertices with an arbitrary orientation of its edges.

Stein's conjecture. If

$$\delta^{0}(D)>k/2,$$

then $D$ contains every oriented path of length $k$.

This is the explicitly labelled conjecture introduced after earlier results on long directed paths and cycles. The supplied text gives no resolution status for it.