The monadic dependence characterization of fixed-parameter tractability

Let ${\mathscr C}$ be a hereditary class of graphs. A class is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from the class using a fixed first-order formula. The monadic dependence conjecture. The first-order model checking problem is fixed-parameter tractable on ${\mathscr C}$ if and only if ${\mathscr C}$ is monadically dependent. The hardness direction is known: first-order model checking is AW[$*$]-hard on every hereditary graph class that is not monadically dependent. Tractability is known for nowhere dense, structurally nowhere dense, and monadically stable classes, but the full tractability direction remains open.