[Demailly and Paun][1] proved the following characterization of nef classes on a compact Kahler manifold:

**Theorem 18.13(a).** Let $X$ be a compact Kähler manifold. A $(1,1)$-class $\alpha$ on $X$ is nef if and only if for every irreducible analytic set $Z \subset X$ of dimension $p$ and every Kähler class $\omega$ we have
$$
\int_Z \alpha \cup \omega^{p-1} \geq 0.
$$

Does the same characterize Kähler classes $\alpha$ if we require the inequality to be strict for all analytic sets and Kähler classes?


  [1]: https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth_book.pdf "Analytic methods in algebraic geometry"