Strong Bogomolov–Gieseker inequality for tilt-semistable objects

Let $X$ be the smooth projective threefold under consideration, let $H$ be its ample divisor, and let $u_{b,w}$ and $\mathcal A(b)$ be the tilt slope and heart defined above, with $w>\frac{b^2}{2}$. For an object $E\in\mathcal A(b)$, write $\operatorname{ch}_i^{bH}(E)$ for its $bH$-twisted Chern character. Strong Bogomolov–Gieseker conjecture. For every $\nu_{b,w}$-semistable $E\in\mathcal A(b)$ satisfying

$$\operatorname{ch}_2^{bH}(E).H =\left(w-\frac{b^2}{2}\right)\operatorname{ch}_0(E)H^3,$$

the inequality

$$\operatorname{ch}_3^{bH}(E)\leq\left(\frac{w}{3}-\frac{b^2}{6}\right)\operatorname{ch}_1^{bH}(E).H^2$$

holds. Although the inequality is known not to hold for all classes on all threefolds, the paper considers whether it holds for the special classes needed for its main theorem.