Kernel inclusion for the WKB-deformed central charge

Let $\Sigma$ be the spectral curve, let $\widetilde{\Sigma}$ be the covering on which the Seiberg–Witten differential is defined, and let

$$\Gamma:= H_1(\widetilde{\Sigma},\mathbb{Z}) / \operatorname{ker} Z$$

be the physical charge lattice, where $Z$ is the classical central-charge homomorphism. Let $Z_\hbar$ be the $\hbar$-deformation defined by the WKB quantum periods:

$$Z_\hbar \in \operatorname{Hom}(H_1(\widetilde{\Sigma},\mathbb{Z}),\mathbb{C}[[\hbar]]).$$

Kernel inclusion conjecture. There is an inclusion of kernels

$$\operatorname{ker} Z \subseteq \operatorname{ker} Z_\hbar.$$

If true, the WKB periods descend from $H_1(\widetilde{\Sigma},\mathbb{Z})$ to a well-defined map on the physical charge lattice, so the quantum deformation preserves the classical equivalence relation. The supplied text does not state whether this inclusion is proved or remains open.