Bang-Jensen–Jordán conjecture for semicomplete digraphs

A digraph is semicomplete if it has no pair of non-adjacent vertices. A tournament is an orientation of a complete graph, hence a semicomplete digraph with no directed 2-cycles. A digraph is $k$-strong if it has at least $k+1$ vertices and remains strongly connected after the deletion of any set of at most $k-1$ vertices. Bang-Jensen–Jordán conjecture. Every $(2k-1)$-strong semicomplete digraph on at least $2k+1$ vertices contains a spanning $k$-strong tournament. The conjecture is motivated by the known case that every 3-strong semicomplete digraph on at least 5 vertices contains a spanning 2-strong tournament; the source does not state a resolution of the general conjecture.