Extension of the \mathscr{Q}-top recursion theorem to arbitrary spectral curves

Let $\Sigma$ be the spectral curve appearing in the $\mathscr{Q}$-top recursion, and let $\omega_{g,n+1}$ denote the multidifferentials produced by that recursion. For $\Sigma=\mathbb{P}^1$, Theorem

$$asserts that these multidifferentials are well-defined and satisfy the stated symmetry, pole, and dilaton properties. **Extension conjecture.** Theorem$$

holds for any $\Sigma$.

The preceding result is proved only when $\Sigma=\mathbb{P}^1$; extending the well-definedness and structural properties of the recursion to general spectral curves is part of the unresolved deformation and quantisation problem.