The forbidden-subgraph characterization conjecture for dominating induced matchings

Let $M$ be a finite set of graphs. Let $\cal S$ be the class of graphs whose connected components are graphs $S_{i,j,k}$. An $M$-free graph is a graph containing no induced subgraph isomorphic to a graph in $M$.

Forbidden-subgraph characterization conjecture. Unless $P=NP$, the dominating induced matching problem is polynomial-time solvable in the class of $M$-free graphs if and only if $M$ contains a graph from $\cal S$.

This conjecture proposes that the known necessary condition for polynomial-time solvability on finitely characterized graph classes is also sufficient. The parser marks the conjecture as disproved, although the supplied surrounding text only says that proving or disproving it is challenging; the resolution evidence should therefore be checked.