Bounded-rank characterization by bounded alternation rank

Let $C$ be a hereditary graph class. Bounded-rank alternation conjecture. The class $C$ has bounded rank if and only if there is $k\in\mathbb{N}$ such that every first-order formula is equivalent on $C$ to a formula of alternation rank $k$. This conjecture generalizes the theorem preceding it in the source and relates the structural notion of rank to a uniform bound on quantifier alternation; the source leaves it as a conjecture for future work.