Extension of refined loop-equation existence to hyperelliptic refined spectral curves

Let $\mathcal{S}_{\bm\mu,\bm\kappa}$ be a hyperelliptic refined spectral curve, and let $\omega_{g,n}$ be the multidifferentials constructed by refined topological recursion. Existence extension conjecture. Theorem 3.2, asserting existence and construction by refined topological recursion of a unique solution of the refined loop equations for all $2g-2+n>0$ when $\Sigma=\mathbb{P}^1$, should extend to any hyperelliptic refined spectral curve.