The codimension-one-nef formulation of the numerical-dimension conjecture

Let $X$ be a projective manifold, let $C\subset X$ be an irreducible curve with ample normal sheaf, and let $L$ be an $\mathbb{R}$-divisor which is nef in codimension $1$.

Codimension-one-nef formulation. If

$$L\cdot C=0,$$

then

$$L\equiv 0.$$

The source presents this as an equivalent formulation of the conjecture that a pseudo-effective line bundle orthogonal to a curve with ample normal sheaf has numerical dimension zero, via divisorial Zariski decomposition. It remains open in general.