Stable-pair integrality and symmetry conjecture for logarithmic Calabi–Yau 3-folds

Let $\bold X$ be a logarithmic Calabi–Yau 3-fold, let $D^{[\beta]}$ be the product of the relevant Hilbert schemes of points on the interiors of the boundary divisors, and let

$$Z_{PT}(\bold X)_\beta=\sum_\chi ev_*[PT_{\beta,\chi}]q^{\chi-\beta\cdot D/2}$$

be the stable-pair partition function, viewed as a lagrangian cycle. Stable-pair integrality conjecture. The element $Z_{PT}(\bold X)_\beta$ is a finite sum of lagrangian cycles whose coefficients are Laurent expansions of rational functions in $q^{\frac 12}$. Each coefficient is invariant under either $q^{\frac 12}\mapsto q^{-\frac 12}$ or $q^{\frac 12}\mapsto -q^{\frac 12}$. This sharpens earlier stable-pair integrality and logarithmic Gromov–Witten/Donaldson–Thomas conjectures. The source gives no resolution, so the assertion remains open.