Convergence conjecture for the genus-deletion ratio

Fix $\theta\in(0,1/2)$, let $g_n$ be the genus sequence used for the high-genus quadrangulations, and let $Q(n,g)$ denote the number of quadrangulations with $n$ faces and genus $g$. Convergence conjecture. There exists a function $r(\theta)$ such that

$$\frac{Q(n,g_n)}{n^2Q(n,g_n-1)}\to r(\theta)$$

as $n\to\infty$. The paper proves that this ratio is of order $\Theta(1)$ in the stated regime and conjectures convergence to a limiting function; the existence and identification of that limit remain open.