A counterexample to a conjecture due to Douglas, Reinbacher and Yau

In “Branes, Bundles and Attractors: Bogomolov and Beyond”, by Douglas, Reinbacher and Yau, the authors state the following conjecture: Consider a simply connected surface XX with ample or trivial canonical line bundle. Then, the Chern classes of any stable vector bundle with rank r≥2r≥2 satisfy 2rc2−(r−1)c12−r212c2(X)≥0. The goal of this short note is to provide two sources of counterexamples to this strong version of the Bogomolov inequality.