Induced subgraph isomorphism hardness conjecture for bipartite permutation graphs

Let $\mathcal X$ be a hereditary subclass of bipartite permutation graphs. The induced subgraph isomorphism problem asks whether an input graph occurs as an induced subgraph of another input graph.

Induced subgraph isomorphism hardness conjecture. Unless $\mathrm{P}=\mathrm{NP}$, the induced subgraph isomorphism problem is NP-hard in $\mathcal X$ if and only if $\mathcal X$ contains all linear forests. Equivalently, for every fixed $k$, the problem can be solved in polynomial time for $P_k$-free bipartite permutation graphs.

The conjecture would characterize the hereditary subclasses of bipartite permutation graphs on which induced subgraph isomorphism is NP-hard, with linear forests as the unique minimal hard class.