The Hodge-Riemann-to-Bogomolov pair conjecture

Let $X$ be a compact complex manifold, and let $(\eta_{d-1},\eta_{d-2})$ be a pair of cohomology classes. A pair is Hodge-Riemann if it satisfies the Hodge-Riemann relations, and it is a Bogomolov pair if every torsion-free sheaf semistable with respect to $\eta_{d-1}$ satisfies the corresponding discriminant inequality against $\eta_{d-2}$. Let $\operatorname{Amp}^{d-1}(X)$ denote the relevant ample cone and $\mathcal K^{d-1}(X)$ the relevant Kähler cone. Hodge-Riemann-to-Bogomolov pair conjecture. If $(\eta_{d-1},\eta_{d-2})$ is a Hodge-Riemann pair and

$$\eta_{d-1}\in \operatorname{Amp}^{d-1}(X)\quad\text{or}\quad\eta_{d-1}\in \mathcal K^{d-1}(X),$$

then $(\eta_{d-1},\eta_{d-2})$ is a Bogomolov pair. This specializes the general conjecture to Hodge-Riemann pairs whose first class lies in the ample or Kähler cone; the supplied text gives no resolution status beyond presenting the assertion in a conjecture environment.