Linear minimum-out-degree conjecture for transitive tournament immersions

Let $D$ be a digraph, let $k$ be a positive integer, and write $\delta^{+}(D)$ for its minimum out-degree. A transitive tournament on $k$ vertices is the tournament whose vertices admit an ordering in which every edge points forward; an immersion replaces its directed edges by pairwise edge-disjoint directed paths. Linear out-degree immersion conjecture. There exists an absolute constant $C>0$ such that, for every positive integer $k$, every digraph $D$ satisfying

$$\delta^{+}(D)\geq Ck$$

contains an immersion of a transitive tournament on $k$ vertices. Lochet proved the analogous assertion with an unspecified function of $k$, while the linear bound for arbitrary digraphs remains open.