Abundance conjecture for numerical dimension equals to zero

At the talk on October 5th the abundance conjecture for a canonical divisor on minimal models was reduced to coincidence of the numerical dimension and the Kodaira dimension for the divisor.
Following Kawamata's paper (2013), I will show that if the numerical dimension for a pair $(X,K_X+B)$ is zero, then the Kodaira dimension for the pair is zero. Here $X$ is a smooth projective variety, and the pair $(X,K_X+B)$ has log-canonical singularities. Thus, the solution of the problem is given in terms of the log-resolutions of minimal models.

The proof is based on the paper by Simpson (1993).