The induced-subdivision dichotomy conjecture for digraphs

Let $D$ be a digraph. For a digraph $G$, let $i_D$ denote the problem of deciding whether $G$ contains an induced subdivision of $D$. A spider is a tree obtained from disjoint directed paths by identifying one end of each path into a vertex, and $C_2$ is the digraph on two vertices with both possible arcs.

Induced-subdivision dichotomy conjecture. $i_D$ is NP-complete unless $D$ is the disjoint union of spiders and at most one $2$-cycle.

This conjecture proposes the complexity dichotomy for induced-subdivision detection in digraphs; the preceding results establish the analogous statement for oriented graphs, while the general digraph case remains open.