Miyaoka-type Chern class inequalities for generically nef sheaves

Let $X$ be a compact Kähler manifold, let $\mathcal{E}$ be a torsion-free coherent sheaf of rank $r$, and let $\alpha_1,\ldots,\alpha_{n-2}$ be Kähler classes on $X$. A sheaf is generically nef with respect to these classes when its restriction to a sufficiently general complete-intersection curve has nef quotient behavior in the sense used in the source. Miyaoka's conjecture. (1) If $c_1(\mathcal{E})$ is nef and $\mathcal{E}$ is generically nef, then

$$c_2(\mathcal{E})\alpha_1\cdots\alpha_{n-2}\geq 0.$$

(2) If $\beta$ is a nef class and $\mathcal{E}$ is $\alpha_1\cdots\alpha_{n-2}\beta$-semistable, then

$$\left(c_2(\mathcal{E})-\frac{r-1}{2r}c_1(\mathcal{E})^2\right)\alpha_1\cdots\alpha_{n-2}\geq 0.$$

Part (2) implies part (1) by the argument cited in the source. The conjecture is solved when $X$ is projective and all the classes involved lie in the Néron–Severi group, but remains open in the general compact Kähler setting.