Genus-independent topological-recursion formula for bipartite quadrangulations

Let $E=e\,\mathrm{id}_N$ be the external matrix of the quartic Kontsevich model, and let $\mathcal{Q}_k^{(g)}$ denote the number of bipartite rooted quadrangulations with $k$ faces on a genus-$g$ surface. Define the spectral data by

$$x(z)=z-\frac{\lambda}{N}\frac{\varrho}{z+\varepsilon},\qquad y(z)=-x(-z),\qquad B(z_1,z_2)=\frac{dz_1dz_2}{(z_1-z_2)^2},$$ $$\varepsilon(e,\lambda)=\frac{1}{6}\left(4e+\sqrt{4e^2+12\lambda}\right),\qquad \varrho(e,\lambda)=\frac{N}{18\lambda}\left(2e\sqrt{4e^2+12\lambda}-4e^2+12\lambda\right).$$

Bipartite quadrangulation conjecture. For $g>0$, the topological recursion for this spectral curve gives

$$\frac{\omega^{TR}_{g,1}(\varepsilon)}{dR(\varepsilon)}=\frac{1}{R'(\varepsilon)}\operatorname{Res}_{q\to-\varepsilon\pm i\sqrt{\lambda\varrho}}\left(K_{\pm}(q,\sigma(q),\varepsilon)\,\omega_{g-1,2}(q,\sigma(q))\right)=\sum_{k=0}^{\infty}\mathcal{Q}_k^{(g)}\frac{(-\lambda)^k}{(2e)^k},$$

where $\sigma(q)=\varepsilon+\frac{\lambda}{N}\frac{\varrho}{q+\varepsilon}$ is the global Galois involution. The conjecture extends the observed genus-one and genus-two computations to arbitrary genus; the paper notes that it was checked for $g=2$ by computer algebra, while a general proof is not provided.