Nakai-Moishezon Criterion for effective $k$-cycles instead of only integral subschemes

The Nakai-Moishezon Criterion states that a Cartier divisor $L$ on a proper scheme over a field is ample if and only if $L^{\dim(Z)} \cdot Z > 0$, for every closed integral subscheme $Z \subset X$ of dimension $k$.

I wondering if, under the same hypothesis (maybe more restrictive), an ample divisor $L$ on $X$ satisfies $L^{k} \cdot \beta > 0$, for every $k$-cycle $\beta \in \overline{NE}_{k}(X)_{\mathbb{R}} \setminus \{ 0 \}$ (the closure of the set of effective $k$-cycles), as well as Kleiman Criterion. Who knows?