The bounded-expansion characterization by poset dimension

Let $\mathcal{C}$ be a monotone class of graphs. For each fixed $h\geq 1$, consider posets of height at most $h$ whose cover graphs belong to $\mathcal{C}$. Bounded-expansion characterization. The class $\mathcal{C}$ has bounded expansion if and only if these posets have bounded dimension for every fixed $h$. This conjecture proposes that bounded expansion is exactly the graph-class condition governing dimension bounds for bounded-height posets; the forward direction follows from the results discussed in the paper, while the backward direction is identified as an open problem.