Asymptotic enumeration conjecture for bipartite maps with prescribed face degrees

Let $(\mathbf{f}^{n})_{n \geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\mathbf{f}^{n},g_n)\geq 2$ for all $n\geq 1$. Assume that $|\mathbf{f}^n|\to+\infty$, that

$$\frac{f^n_j}{|\mathbf{f}^n|}\to\alpha_j\quad\text{for all }j\geq 1,$$

where $\sum_{j\geq 1}j\alpha_j=1$, that $g_n/|\mathbf{f}^n|\to\theta$ with $0\leq\theta\leq\frac{1}{2}\sum_{j\geq 1}(j-1)\alpha_j$, and that $\sum_{j\geq 1}j^2\alpha_j<+\infty$. Asymptotic enumeration conjecture. There exists a function $\varphi$ such that

$$\beta_{g_n}(\mathbf{f}^{n})=|\mathbf{f}^n|^{2g_n}\exp\left(\varphi\left(\theta,(\alpha_j)_{j\geq 1}\right)|\mathbf{f}^n|+o\left(|\mathbf{f}^n|\right)\right).$$

This conjecture predicts precise exponential-scale asymptotics for the number of bipartite maps of genus $g_n$ with prescribed face degrees. It is presented as a belief following a related ratio-convergence result, and no resolution is supplied in the source.