Fixed-parameter tractability of first-order model checking on monadically dependent classes

A graph class $\mathscr{C}$ is monadically dependent if it has the model-theoretic property described in the survey. For a graph $G$, let $n$ denote its size, and let $\mathsf{FO}$ denote first-order logic.

The monadic dependence model-checking conjecture. For every monadically dependent class $\mathscr{C}$ there is a constant $c\in\mathbb N$ and an algorithm that, given a graph $G\in\mathscr{C}$ and an $\mathsf{FO}$ sentence $\varphi$, decides whether $G\models\varphi$ in time $\mathcal{O}_{\mathscr{C},\varphi}(n^c)$.

This conjecture would extend the fixed-parameter tractability of first-order model checking from nowhere dense classes to all monadically dependent classes. The survey identifies it as a main algorithmic goal and states that it remains open.