Nowhere-dense characterization of modeling limits

Let $\mathcal C$ be a monotone class of finite graphs, meaning that it is closed under taking subgraphs. A class is nowhere dense in the standard sparse-graph sense. The class $\mathcal C$ admits modeling limits if every FO-convergent sequence of graphs from $\mathcal C$ has a modeling limit. Nowhere-dense modeling-limit conjecture.

$$\mathcal C\text{ admits modeling limits if and only if }\mathcal C\text{ is nowhere dense}.$$

This gives an exact proposed characterization of the monotone graph classes supporting modeling limits. In the supplied text, the statement is subsequently obtained as a corollary, so its status is solved.