Douglas–Reinbacher–Yau strong Bogomolov inequality for simply connected surfaces

Let $S$ be a simply connected smooth projective algebraic surface with trivial or ample canonical bundle. For a stable vector bundle of rank $r$ on $S$, with Chern classes $c_1$ and $c_2$, and with nontrivial moduli space, the discriminant is

$$2rc_2-(r-1)c_1^2.$$

Douglas–Reinbacher–Yau conjecture. The Chern classes should satisfy the improved Bogomolov inequality

$$2rc_2-(r-1)c_1^2-\frac{r^2}{12}c_2(S)\geq 0.$$

The conjecture was proposed in connection with the Strong Bogomolov Inequality. It is false, as proved in the cited work of Coskun and of Nakashima and collaborators; consequently the proposed inequality does not hold in the stated generality.