In the complex case, it is not always possible to find a smooth semipositive form in a nef class, so the image of $\mathcal{Z}^{1,1}_{+}(X)$ in $N^{1}(X)$ is strictly contained in $\Nef(X)$ in general, see [DPS94, Example 1.7]. In the non-Archimedean setting, the situation is unclear.