Linear out-degree bound for transitive tournament immersion in digraphs

Ascertain whether there exists a constant c > 0 such that, for every integer k ≥ 1, every digraph with minimum out-degree at least c · k contains an immersion of the transitive tournament on k vertices.

Background

Lochet showed that high minimum in/out-degrees do not guarantee immersion of K_3 in general digraphs, but established polynomial out-degree bounds for immersing transitive tournaments. The question asks whether linear out-degree suffices, paralleling linear bounds known for undirected clique immersions in special classes.