Generalized abundance conjecture for projective smooth pairs

Let $X$ be a smooth projective variety and let $E$ be a simple normal crossing divisor on $X$. Let $\kappa_\sigma$ denote Nakayama's numerical dimension. Generalized abundance conjecture. If $K_X+E$ is the log canonical divisor of this smooth pair, then

$$\kappa (X, K_X+E)=\kappa_\sigma(X, K_X+E).$$

This conjecture is stated to be still widely open and is described as one of the most important conjectures in the theory of minimal models.