Equality of the non-nef locus and the restricted base locus on normal projective varieties

Let $X$ be a normal projective variety, let $L$ be a big line bundle on $X$, and let $x\in X$ be a point such that for every divisorial valuation $\nu$ centered at $x$ there is an infinite sequence $\sigma_k\in H^0(kL)$ satisfying $\nu(\sigma_k)=o(k)$.

Non-nef locus conjecture. There exists an ample divisor $A$ and an infinite sequence $\tau_k\in H^0(kL+A)$ such that $\tau_k(x)\neq 0$.

This criterion is presented as equivalent to the equality $\operatorname{NNef}(D)=\mathbb B_-(D)$ for all pseudoeffective divisors on normal projective varieties. The equality is known for smooth varieties and, in general, the non-nef locus is contained in the restricted base locus; the extension to arbitrary normal projective varieties is the issue under consideration.