The Bogomolov inequality for Hodge-Riemann pairs

Let $X$ be a compact complex manifold of dimension $d$, and let $(\eta_{d-1},\eta_{d-2})$ be a pair of cohomology classes of bidegrees $(d-1,d-1)$ and $(d-2,d-2)$, respectively. A pair is Hodge-Riemann when it satisfies the Hodge-Riemann relations. For a torsion-free coherent sheaf $E$ on $X$, let $r=\operatorname{rk}(E)$ and define its discriminant by

$$\Delta(E)=2r c_2(E)-(r-1)c_1(E)^2.$$

Assume that $\eta_{d-1}$ defines slope semistability. Bogomolov inequality for Hodge-Riemann pairs. If $(\eta_{d-1},\eta_{d-2})$ is a Hodge-Riemann pair, then every $\eta_{d-1}$-semistable torsion-free sheaf $E$ satisfies

$$\int_X \bigl(2r c_2(E)-(r-1)c_1(E)^2\bigr)\cdot\eta_{d-2}\ge 0.$$

This is the proposed generalization of the classical Bogomolov inequality from powers of an ample class to arbitrary Hodge-Riemann pairs. The source presents it as a conjectural statement and proves only various cases.