Growth constant at the logarithmic genus threshold

Let $g$ be a genus function and let $\mathcal A^g$ be one of the labelled graph classes considered in the paper. Logarithmic-threshold growth-constant conjecture. If $c>0$ is a constant and

$$g(n)\sim \frac{cn}{\log n},$$

then $\mathcal A^g$ has a growth constant $\gamma=\gamma(c)$. This concerns the boundary regime where the radius of convergence is positive, but the paper does not establish existence of a growth constant.