Eppstein's integer-power conjecture for planar subgraph counts

Let $H$ be a graph, and let $\operatorname{ex}_{\mathcal{P}}(n,H,\emptyset)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph.

Eppstein's conjecture. For every graph $H$, there exists a non-negative integer $k$ such that

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

This is a restricted planar version of a question posed by Eppstein about subgraph counts in minor-closed graph families. The paper presents it as an open conjecture, while verifying it for trees and establishing general upper bounds.