Extension of the refined variational formula to arbitrary spectral curves

Let $\Sigma$ be a spectral curve, let $\mathcal{S}_{\bm\kappa,\bm\mu}(\bm t)$ satisfy the refined deformation condition with respect to the parameters $t_l$, and let $\omega_{g,n+1}$ and $F_g$ be the multidifferentials and free energies of the refined recursion. When $\Sigma=\mathbb{P}^1$, Theorem

gives the variational formulas for $F_g$ and $\omega_{g,n+1}$. **Variational extension conjecture.** Theorem

holds for any $\Sigma$.

The displayed variational formula is proved in the paper for $\Sigma=\mathbb{P}^1$. Its extension to general spectral curves is therefore left open by the source.