Integer-power conjecture for forbidden-subgraph planar counts

Let $\mathcal{F}$ be a finite set of graphs, let $H$ be a graph, and let $\operatorname{ex}_{\mathcal{P}}(n,H,\mathcal{F})$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph containing no graph in $\mathcal{F}$ as a subgraph.

Integer-power conjecture. For all finite sets of graphs $\mathcal{F}$ and all graphs $H$, there is an integer $k$ such that

$$\operatorname{ex}_{\mathcal{P}}(n,H,\mathcal{F})=\Theta(n^k).$$

This generalizes the preceding planar conjecture to forbidden subgraphs. The paper poses it as open; its precise scope includes arbitrary finite forbidden families and arbitrary graphs being counted.