On the fixed and negative part of a linear system

Question

Let $X$ and $Z$ be smooth complex projective varieties and let $f:X\rightarrow Z$ be a contraction (i.e. $f_\ast\mathcal{O}_X=\mathcal{O}_Z$). Let $F$ be an effective $\mathbb{R}$-divisor on $X$ such that $F$ is the only element in its relative linear system $|F/Z|_{\mathbb{R}}$. Suppose furthermore that $F=F_1+F_2$ where the $F_i$ are effective with no common components and $F_2=N_\sigma(F_2/Z)$. Let now $0\leq \epsilon \ll 1$: is it true that $N_\sigma(F+\epsilon F_2/Z)$ contains $F_2$ in its support?