Demailly and Paun 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?

Demailly and Paun 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?