Pilipczuk–Toruńczyk's rank conjecture for elementarily-fpt model checking

Let $C$ be a hereditary graph class. For $d\in\mathbb{N}$, let $T_d$ denote the class of all trees of depth $d$. Pilipczuk–Toruńczyk's rank conjecture. The class $C$ has elementarily fixed-parameter tractable model checking if and only if there is some $d\in\mathbb{N}$ such that $C$ does not transduce $T_d$. The conjecture would characterize hereditary classes with elementarily-fpt first-order model checking by bounded rank. The source notes that the conjecture motivates the paper and remains a subject of future work; its status is not resolved there.