Fixed-parameter tractability conjecture for first-order model checking on hereditary classes

Let $\mathscr C$ be a hereditary class of structures. A class is monadically NIP if it remains NIP after arbitrary unary predicates are added. Hereditary first-order model-checking conjecture. First-order model checking is fixed-parameter tractable on $\mathscr C$ if and only if $\mathscr C$ is monadically NIP. Here fixed-parameter tractability means an algorithm with running time $f(\varphi)\cdot |G|^c$ for some computable function $f$ and constant $c$. The conjecture would characterize exactly the hereditary classes admitting tractable first-order model checking; the paper notes that all currently known tractable hereditary classes are monadically NIP, while the converse is not established in general.