Douglas–Reinbacher–Yau strengthened Bogomolov inequality

Let $X$ be a non-singular, simply-connected, compact Kähler manifold of dimension $n$, with Kähler class $H$, and assume that $X$ has trivial or ample canonical bundle. Let $E$ be an $H$-stable holomorphic vector bundle on $X$ of rank $r\geq 2$. Define

$$\Delta(E)=\frac{1}{r^2}\left(2r c_2(E)-(r-1)c_1(E)^2\right)\cdot H^{n-2}.$$

Douglas–Reinbacher–Yau strengthened Bogomolov conjecture. The Chern classes of $E$ satisfy

$$\Delta(E)\geq \frac{1}{12}c_2(TX)\cdot H^{n-2}.$$

This is a proposed strengthening of the usual Bogomolov inequality for stable bundles, motivated by physical considerations and applicable to manifolds with trivial or ample canonical bundle. The supplied text does not state whether the conjecture has been resolved.