Bogomolov–Gieseker inequality in the reduced parameter range

Let $X$ be the canonical line bundle of $\mathbb{P}^3$, with projection $\pi:X\to\mathbb{P}^3$, and let $H$ be the pullback of the hyperplane class. Let $\mathrm{Coh}_{0}^{\beta}(X)$ be the tilted heart and let $\nu^{\beta,\alpha}$ denote the tilt slope. Reduced Bogomolov–Gieseker conjecture. For rational $\beta,\alpha$ satisfying

$$\alpha>\frac{1}{2}\beta^2,\qquad -\frac{1}{2}\leqslant\beta\leqslant0,\qquad \sqrt{2\alpha-\beta^2}<\frac{1}{2},$$

and every $\nu^{\beta,\alpha}$-semistable object $E\in\mathrm{Coh}_{0}^{\beta}(X)$ with tilt-slope $\beta$, one has

$$\mathrm{ch}_{3}(\pi_*E)\leqslant\frac{2\alpha-\beta^2}{6}H^2\mathrm{ch}_{1}(\pi_*E).$$

The source presents this as a reduction equivalent to the preceding Bogomolov–Gieseker conjecture, so it is not a separate mathematical claim, but records the parameter range to which the problem can be reduced.