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?
This is false. There exists a smooth projective surface $X$ with a strictly nef divisor $D$ (so $D\cdot C>0$ for all curves $C\subset X$) and yet $(D^2)=0$, so in particular $D$ is not ample, see e.g. here.
Letting $\alpha=c_1(D)$ we have $\int_C \alpha>0$ for all curves $C$. Since $\alpha$ is nef, for any Kähler form $\omega$ we have $\int_X \alpha\wedge\omega\geq 0.$ And if this integral was equal to zero, this would force $\alpha$ to be the zero class (by the Hodge index theorem), which is not the case.
