The weak DRY conjecture for stable bundles on Calabi–Yau threefolds

Let $X$ be a Calabi–Yau threefold with $pi_1(X)=0$. For $c\in H^4(X,\mathbf{Z})$, call $c$ a DRY class if there exist an ample class $H\in H^2(X,\mathbf{R})$ and an integer $\mathcal{N}$ such that

$$c=\mathcal{N}\left(H^2+\frac{c_2(X)}{24}\right).$$

Call $c$ a Chern class if there exists a stable $SU(\mathcal{N})$ vector bundle $V$ on $X$ with $c=c_2(V)$. Weak DRY conjecture. Every DRY class $c\in H^4(X,\mathbf{Z})$ is a Chern class.

The conjecture gives a sufficient condition for a cohomology class to occur as the second Chern class of a stable vector bundle with trivial first Chern class. The paper proves it for all ranks $\mathcal{N}\geq 4$ in the stated elliptically fibered cases, apart from finitely many exceptions, while the general assertion remains open.