Bang-Jensen–DeVos–Mütze conjecture on strong tournaments in semicomplete digraphs

Let $D$ be a semicomplete digraph, meaning that for every pair of distinct vertices, at least one of the two possible directed arcs is present. A digraph is $k$-strong if it has at least $k+1$ vertices and remains strongly connected after the deletion of any subset of $k-1$ vertices. A tournament is an oriented complete graph, and a spanning subdigraph contains all vertices of $D$.

Bang-Jensen–DeVos–Mütze conjecture. Every $(2k-1)$-strong semicomplete digraph on at least $2k+1$ vertices contains a spanning $k$-strong tournament.

The source presents this as a conjecture supported by its arc-connectivity analogue, while also giving sharpness results for the order and connectivity bounds and a counterexample for a broader class of almost semicomplete digraphs. Its status is not resolved in the supplied text.