Monotonicity conjecture for induced-subdivision detection

Let $D$ and $D'$ be digraphs. For a digraph $G$, let $i_D$ (respectively, $i'_D$) denote the corresponding induced-subdivision detection problem. Say that $D$ is an induced subdigraph of $D'$ if $D'$ contains a vertex-induced copy of $D$.

Monotonicity conjecture. If $i_D$ (respectively, $i'_D$) is NP-complete, then for every digraph $D'$ containing $D$ as an induced subdigraph, $i_{D'}$ (respectively, $i'_{D'}$) is NP-complete.

The conjecture is proposed as a tool for proving a full complexity dichotomy. It covers both variants discussed in the paper and is open according to the surrounding remarks.