The generalized Bogomolov–Gieseker conjecture for the quadratic-quartic threefold

Let $X$ be a smooth projective threefold that is the complete intersection of a quadratic and a quartic hypersurface in $\mathbb{P}^5$, and let $H$ be an ample class. For $B=\beta H$, write $\overline{\Delta}_H(E)=\left(H^2\mathop{\mathrm{ch}}\nolimits_1(E)\right)^2-2H^3\mathop{\mathrm{ch}}\nolimits_0(E)\cdot H\mathop{\mathrm{ch}}\nolimits_2(E)$, and define the twisted Chern characters $\mathop{\mathrm{ch}}\nolimits_i^{\beta H}(E)$ in the usual way. Assume that $E$ is $\nu_{\alpha,\beta,H}$-tilt semistable for some $\alpha>\frac{1}{2}\beta^2$. Generalized Bogomolov–Gieseker conjecture. One should have

$$Q_{\alpha,\beta}(E):=\left(2\alpha-\beta^2\right)\overline{\Delta}_H(E)+4\left(H\mathop{\mathrm{ch}}\nolimits_2^{\beta H}(E)\right)^2-6H^2\mathop{\mathrm{ch}}\nolimits_1^{\beta H}(E)\mathop{\mathrm{ch}}\nolimits_3^{\beta H}(E)\geq0.$$

This is the conjectural Bogomolov–Gieseker inequality for tilt-stable objects on this Calabi–Yau threefold; proving it yields the expected quadratic inequality needed for constructing Bridgeland stability conditions. The paper’s stated goal is to establish it in the specified complete-intersection case, while the parser supplies no evidence that the conjecture has been resolved, so its database status remains open.