Bogomolov–Gieseker type conjecture for tilt-stable complexes

Let $X$ be the smooth projective threefold under consideration, let $H$ be its ample divisor, and let $\beta\fa$ be a real number. For an object $E\in\operatorname{Coh}^{\beta}(X)$ that is $\nu_{\alpha,\beta}$-semistable, write

$$\overline{\Delta}^{\beta}_H(E)=(H^2\operatorname{ch}^{\beta}_1(E))^2-2H^3\operatorname{ch}^{\beta}_0(E)\cdot(H\operatorname{ch}^{\beta}_2(E)).$$

Bogomolov–Gieseker type conjecture. One should have

$$\alpha^2\overline{\Delta}^{\beta}_H(E)+4\left(H\operatorname{ch}^{\beta}_2(E)\right)^2-6H^2\operatorname{ch}^{\beta}_1(E)\operatorname{ch}^{\beta}_3(E)\geq0.$$

This is the stronger Bogomolov–Gieseker type inequality proposed for tilt-stable complexes; unlike the preceding discriminant inequality, its status is not resolved by the supplied source context.