Functional relations under symplectic exchange for topological-recursion differentials

Let $\mathcal{C}$ be a compact Riemann surface. Let $x$ and $w$ be meromorphic functions on $\mathcal{C}$ such that $\mathrm{d}x$ and $\mathrm{d}w$ have no common zeroes, and let $\mathcal{B}$ be a fundamental bidifferential of the second kind. Denote by $\omega_{g,n}$ the differentials obtained by topological recursion from $(\mathcal{C},x,w,\mathcal{B})$, and by $\omega_{g,n}^{\vee}$ those obtained from $(\mathcal{C},w,x,\mathcal{B})$. Define

$$\widetilde{\omega}_{0,2}=\widetilde{\omega}_{0,2}^{\vee}=\mathcal{B}.$$

Functional-relations conjecture. For every $2g-2+n\geq 0$, these differentials satisfy the functional relations of Theorem~, after those relations are expressed as relations between meromorphic differentials on $\mathcal{C}$. This is a proposed general relation between topological recursion and the symplectic exchange $x\mapsto w$. The supplied material does not provide a resolution status or explain the precise functional relations beyond the cited theorem.