The numerical-dimension-zero abundance conjecture

Work over $\mathbb{C}$. Let $(X,\Delta)$ be a projective klt log pair of dimension $n\geq 2$. Let $\kappa(X,\Delta)$ denote the Kodaira dimension of $K_X+\Delta$, and let $\nu(K_X+\Delta)$ denote its numerical dimension. The numerical-dimension-zero abundance conjecture. If

$$\kappa(X,\Delta)=0,$$

then

$$\nu(K_X+\Delta)=0.$$

This conjecture is a reduction of the abundance conjecture to the case of Kodaira dimension zero. The source describes this implication as well known, but the conjecture itself is not resolved in the supplied text.