Generalized abundance conjecture for dlt pairs

Let $(X,\Delta)$ be a $\mathbb Q$-factorial projective dlt pair. The numerical Kodaira dimension $\kappa_\sigma(X,K_X+\Delta)$ and the invariant Kodaira dimension $\kappa_\iota(X,K_X+\Delta)$ are the two dimensions appearing in generalized abundance. Generalized abundance conjecture. One has

$$\kappa_\sigma(X,K_X+\Delta)=\kappa_\iota(X,K_X+\Delta).$$

For a $\mathbb Q$-divisor, $\kappa_\iota$ agrees with the usual Iitaka dimension. The conjecture is presented as a central higher-dimensional conjecture and, in the paper, provides the reduction of logarithmic subadditivity; it remains open in the stated generality.