Refined BPS formula for hypergeometric-type spectral curves

Let $\mathcal{S}^{\bm\mu}=(\overline{\Sigma},x,y,D(\bm\mu))$ be a refined spectral curve of hypergeometric type. Let $(\Gamma,Z,\Omega)$ be its corresponding refined BPS structure, let \lower0.55ex\text{\mathchar'26}\mkern-11.5mu Z be the quantum correction to the central charge, and let $\Omega_n(\gamma)$ denote the coefficient of $t^n$ in $\Omega(\gamma)$. For the double Bernoulli polynomial $B_{2,2g}$, define

$$\mathsf{B}_{2,2g}^{[n]}(\gamma):=B_{2,2g}\left(\frac{\mathscr{Q}}{2}+\frac{\lower0.55ex\text{$\mathchar'26$}\mkern-11.5mu Z(\gamma)}{2\pi i}+n\frac{\mathscr{Q}}{2}\,\Big|\,\beta^{\frac{1}{2}},-\beta^{-\frac{1}{2}}\right).$$

Refined BPS free-energy conjecture. The refined topological recursion free energy satisfies

$$F_g=(-1)^{2g-2}\sum_{\gamma\in\Gamma}\sum_{n\in\mathbb{Z}}\frac{\mathsf{B}_{2,2g}^{[n]}(\gamma)}{4g(2g-1)(2g-2)}\Omega_n(\gamma)\left(\frac{2\pi i}{Z(\gamma)}\right)^{2g-2}.$$

This extends the formula proved in the paper for the Weber, Whittaker, degenerate Bessel, and Airy refined spectral curves to the remaining hypergeometric-type curves. The authors note that technical difficulties arise from second-order poles and that only finite-order computations are currently available in these examples, so the claim remains open.