Jackson–Thomassen conjecture on strong orientations of digraphs

Let $D=(V,A)$ be a digraph. It 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. An oriented subdigraph is obtained by retaining exactly one arc from every directed $2$-cycle of $D$.

Jackson–Thomassen conjecture. Every $2k$-strong digraph has a spanning $k$-strong oriented subdigraph.

For $k=1$, this follows from a result of Boesch and Tindell on strong orientations of mixed graphs, and the conjecture is known for symmetric digraphs by Thomassen. It remains open even for $k=2$ in general.