Integrality conjecture for coarse-moduli BPS functions

Let $X$ be a Calabi–Yau $3$-fold over $\mathbb C$, and let $(\tau,G,\leqslant)$ be a generic Gieseker stability condition. For each $\alpha\in C(\operatorname{coh}(X))$, let $F^\alpha(\tau)\in\operatorname{CF}(\mathcal M_{\rm ss}^\alpha(\tau))$ be the constructible function defined by the displayed formula in the source. Integrality conjecture for $F^\alpha(\tau)$. The functions $F^\alpha(\tau)$ are $\mathbb Z$-valued for all $\alpha\in C(\operatorname{coh}(X))$. This conjecture is presented as implying the BPS integrality conjecture for stability conditions; its resolution is not given in the supplied material.