Linear and quadratic loop equations for the quartic Kontsevich model

Let $R:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be the ramified covering defined by the model, let $\beta_1,\ldots,\beta_{2d}$ be its ramification points, and let $\sigma_i$ be the corresponding local Galois involution near $\beta_i$. Define $\omega_{0,1}(z)=-R(-z)R'(z)\,dz$ and $\omega_{0,2}(u_1,z)=\frac{du_1\,dz}{(u_1-z)^2}+\frac{du_1\,dz}{(u_1+z)^2}$; for $2-2g-m<0$, define $\omega_{g,m}$ by the stated residue system. Loop-equation conjecture. For all $g\geq0$ and $m\geq1$, the meromorphic differentials $\omega_{g,m}$ are symmetric and satisfy, near every $\beta_i$, the linear loop equation

$$\omega_{g,m}(u_1,\ldots,u_{m-1},z)+\omega_{g,m}(u_1,\ldots,u_{m-1},\sigma_i(z))=\mathcal{O}(z-\beta_i)\,dz$$

and the quadratic loop equation

$$\omega_{g-1,m+1}(u_1,\ldots,u_{m-1},z,\sigma_i(z))+\sum_{\substack{I_1\uplus I_2=\{u_1,\ldots,u_{m-1}\}\g_1+g_2=g}}\omega_{g_1,|I_1|+1}(I_1,z)\omega_{g_2,|I_2|+1}(I_2,\sigma_i(z))=\mathcal{O}((z-\beta_i)^2)(dz)^2.$$

The claim is motivated by the verified low-degree cases and would establish the loop-equation structure underlying blobbed topological recursion for the quartic model; the general assertion remains open in the supplied text.