nonnef locus and discrete valuation.

When $X$ is a smooth projective variety and $D$ is a divisor on $X$, it is known that the nonnef locus $\mathbf{B}_-(D)$ is the union of the centers of the discrete valuations $\sigma$ such that $\overline\sigma(D):=\inf${$\sigma(E)|0\leq E\equiv D$}$>0$. But can we say that $\overline\sigma(D)>0$ if and only if Center$(\sigma)$ is contained in the nonnef locus of $D$? (In other words, can there be a discrete valuation $\sigma$ such that $\sigma(D)=0$ and the center of $\sigma$ is still in the nonnef locus of $D$?)

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If $X$ is a smooth projective variety, then $\overline{\sigma}(D)>0$ if and only if $\mathrm{Center}(\sigma)\subseteq \mathbf{B}_-(D)$. A reference is "Asymptotic invariants of base loci" by Ein, Lazarsfeld, Mustata, Nakamaye, Popa, theorem 2.8
Actually they do this for a big divisor, but this should imply the same result for all pseudoeffective divisors, by using that $\mathbf{B}_-(D)$ is the union of the non-nef loci of $D+A_m$, where $A_m$ is a sequence of amples that goes to 0.