Polynomial upper bound for genus-increment growth ratios

Let $\mathcal A^h$ be one of the graph classes considered in the paper, indexed by graphs on $[n]$ embeddable in surfaces of Euler genus at most $h$. Genus-increment growth-ratio conjecture. There are constants $\alpha$ and $\beta$ such that, for every $0\leq h\leq n$,

$$\frac{|\mathcal A^{h+2}_n|}{|\mathcal A^h_n|}\leq \alpha n^\beta.$$

The conjecture is proposed as an upper bound when the allowed Euler genus is increased by two; the surrounding discussion suggests that one might take $\beta=2$, and notes that this would imply a conjecture about the planar growth constant.