Győri–Paulos–Salia–Tompkins–Zamora planar forbidden-subgraph counting conjecture

Let $\mathcal F$ be a finite set of graphs, let $H$ be a graph, and let $\mathcal G$ be the class of all planar graphs with no subgraph isomorphic to any member of $\mathcal F$. Assume that $H\in\mathcal G$. Győri–Paulos–Salia–Tompkins–Zamora's conjecture. There is an integer $k$ such that

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

This conjecture concerns the polynomial order of homomorphism and subgraph counts in planar graph classes with finitely many forbidden subgraphs. The paper notes that its bounded-expansion theorem gives a negative answer to a related question, but does not state a resolution of this conjecture.