Strong Bogomolov–Gieseker inequality for stable sheaves on quintic threefolds

Let $X\subset\mathbb{P}^4$ be a smooth quintic threefold and let $H=c_1(\mathcal{O}_X(1))$. For a torsion-free $H$-slope-stable sheaf $E$ on $X$ with $c_1(E)/\operatorname{rank}(E)=-H/2$, define the discriminant by

$$\Delta(E)=\operatorname{ch}_1(E)^2-2\operatorname{ch}_0(E)\operatorname{ch}_2(E).$$

Strong Bogomolov–Gieseker conjecture. One has

$$\frac{\Delta(E)\cdot H}{\operatorname{rank}(E)^2}>1.5139\cdots.$$

This is a proposed strengthening of the classical Bogomolov–Gieseker inequality for stable sheaves on quintic threefolds. The source proves the conjecture in the rank-two case; the general case remains open.