The abundance conjecture for projective manifolds

Let $X$ be a projective manifold that is not uniruled. A good minimal model is a minimal model $X'$ with terminal singularities and semi-ample canonical bundle.

Abundance conjecture. The manifold $X$ admits a good minimal model $X'$ such that

$$\kappa(X)=\kappa(X').$$

If $K_X$ is nef, then

$$X\simeq X'.$$

The conjecture is known for projective manifolds of complex dimension at most $3$, and is used in the paper as an assumption for its main theorems. It remains open in higher dimensions.