Growth-constant threshold for order-dependent surface classes

Let $g$ be a genus function, and let $\mathcal A^g$ denote one of the graph classes considered in the paper, such as $\mathcal E^g$, $\mathcal O\mathcal E^g$, $\mathcal N\mathcal E^g$, or $\mathcal O\mathcal E^g\cap\mathcal N\mathcal E^g$. Let $\gamma_{\mathcal P}$ be the growth constant of the planar graphs. Growth-constant threshold conjecture. The class $\mathcal A^g$ has growth constant $\gamma_{\mathcal P}$ if and only if

$$g(n)=o\!\left(\frac{n}{\log n}\right).$$

The paper proves the forward necessary condition and the matching conclusion under the stronger hypothesis $g(n)=o(n/\log^3 n)$; the conjecture asks whether the threshold can be improved to $o(n/\log n)$.