The structural conjecture for nowhere dense classes without bounded expansion

Let $\mathcal C$ be a monotone nowhere dense class of graphs. A $p$-subdivision is obtained by replacing every edge of a graph by a path of length $p+1$.

Structural conjecture for nowhere dense classes. If $\mathcal C$ does not have bounded expansion, then there exists an integer $p$ such that $\mathcal C$ includes $p$-subdivisions of graphs with arbitrarily large chromatic number and girth.

This conjecture would provide the structural input needed to extend the paper's characterization of first-order definable colorings from the proved cases to all hereditary addable topologically closed classes. Its status is presented as open in the paper.