The GHM conjecture on the Fredholm determinant and modified grand potential

Let $\rho_S$ be the inverse of the quantized mirror-curve operator acting on $L^2(\mathbb R)$, let $\Xi_S(\kappa,\boldsymbol{\xi},\hbar)$ denote its Fredholm determinant, and write $\kappa={\rm e}^{\mu}$. Let $\mathsf{J}_S(\mu,\boldsymbol{\xi},\hbar)$ be the modified grand potential. GHM conjecture. The Fredholm determinant of $\rho_S$ is given by

$$\Xi_S(\kappa,\boldsymbol{\xi},\hbar)=\sum_{n\in\mathbb Z}\exp\left(\mathsf{J}_S(\mu+2\pi\mathsf{i}n,\boldsymbol{\xi},\hbar)\right).$$

This is the main conjectural relation between the spectral determinant and topological-string data; the supplied source gives no resolution of it.