Campana's abundance-type conjecture for decompositions of the canonical divisor

Let $X$ be a projective manifold. Suppose that, for some positive integer $N$,

$$N K_X=A+B,$$

where $A$ is an effective divisor and $B$ is a pseudo-effective line bundle. Campana's abundance-type conjecture. Then

$$\kappa(X)\geq\kappa(A).$$

The text presents this as a reduction of the refined Kodaira-dimension conjecture and notes that it is sufficient for the intermediate cases under consideration; the supplied text gives no resolution of the conjecture in general. The source also notes that $A$ may, after suitable blow-ups, be assumed spanned.