The bounded-expansion characterization via poset dimension

Let $\mathcal{C}$ be a class of graphs closed under taking subgraphs. For each integer $h\geq 1$, consider posets of height $h$ whose cover graphs belong to $\mathcal{C}$.

Bounded-expansion characterization. The class $\mathcal{C}$ has bounded expansion if and only if, for each $h\geq 1$, the posets of height $h$ whose cover graphs are in $\mathcal{C}$ have bounded dimension.

The theorem proved in the paper establishes one of the two implications; the converse remains open. Thus the conjecture proposes a characterization of bounded-expansion graph classes in terms of uniformly bounded dimension for bounded-height posets with cover graphs in the class.