Minimum outdegree forcing immersions of transitive tournaments

A simple digraph is a directed graph with no parallel edges or loops, and a transitive tournament on $t$ vertices is the orientation of the complete graph in which the vertices can be ordered so that every edge points forward. An immersion maps vertices injectively and represents each directed edge by an edge-disjoint directed path.

Immersion conjecture for transitive tournaments. There exists a function $f$ so that every simple digraph of minimum outdegree $f(t)$ contains an immersion of the transitive tournament on $t$ vertices.

This is a weakening of Mader's subdivision conjecture and is presented in the source as an interesting open problem.