Refined correlator-volume correspondence for bordered Klein surfaces

Let $g\in\frac12\mathbb{Z}_{\geq0}$ and $n\in\mathbb{Z}_{\geq1}$ satisfy $2-2g-n<0$. Let $\omega_{g,n}^{\mathfrak b}$ be the refined correlator on the refined spectral curve, let $V_{g,n}^{\epsilon}(L_{[n]})$ be the total volume of the corresponding regularised moduli space, and let $\hat{\mathcal{L}}_{L}^{-1}$ denote termwise inverse Laplace transformation. Introduce $b$ by $\mathfrak b=-\frac{b}{\sqrt{1+b}}$. Refined correlator-volume conjecture. The relation between the refined correlators and total volumes holds for all such $(g,n)$, namely

$$\prod_{i=1}^nL_i\cdot V_{g,n}^{\epsilon}(L_{[n]})=(1+b)^g\left(\hat{\mathcal{L}}_{L_1}^{-1}\cdots\hat{\mathcal{L}}_{L_n}^{-1}\right)\!\cdot\omega_{g,n}^{\mathfrak b}\big|_{b=1}.$$

This extends the theorem established for the cases with $2-2g-n=-1$, including $(g,n)=(0,3),(\frac12,2),(1,1)$. The conjecture proposes that refined topological recursion recovers all total hyperbolic volumes, but it remains open beyond the tested cases and requires the analytic and convergence properties of the refined correlators.