The dependence conjecture for first-order model checking on hereditary graph classes

Let $\mathscr C$ be a hereditary class of graphs. A class is dependent if it has the model-theoretic non-independence property, and first-order model checking is fixed parameter tractable if, given a graph $G\in\mathscr C$ and a first-order sentence $\varphi$, whether $G\models\varphi$ can be decided in time $f(|\varphi|)\lvert G\rvert^{O(1)}$ for some computable function $f$. Dependence conjecture. Under the standard assumption $\mathrm{FPT}\neq\mathrm{AW}[\ast]$, first-order model checking is fixed parameter tractable on $\mathscr C$ if and only if $\mathscr C$ is dependent. This conjecture proposes a precise correspondence between model-theoretic dependence and the algorithmic tractability of first-order model checking on hereditary graph classes; the stated equivalence is conditional on the standard complexity-theoretic assumption and is not resolved in general.