Non-Kähler nef intersection conjecture

Let $X$ be a compact complex manifold, let $\mathcal{N}\subset H^{1,1}(X,\mathbb{R})$ be its cone of nef $(1,1)$-classes, let $[\alpha]\in\mathcal{N}$, and let $V\subset X$ be a closed irreducible analytic subvariety of dimension $k>0$. Nef intersection conjecture. One has

$$\int_V\alpha^k\geqslant 0.$$

This question concerns the expected nonnegativity of intersection numbers of nef $(1,1)$-classes on compact complex manifolds without a Kählerity assumption; it was raised in 2016 and is presented here as a basic question, with no resolution indicated in the source.