The good minimal model conjecture for klt pairs

Let $(X,\Delta)$ be an $n$-dimensional klt pair. A pair is pseudo-effective when $K_X+\Delta$ is pseudo-effective.

Good minimal model conjecture. If $K_X+\Delta$ is pseudo-effective, then $(X,\Delta)$ has a good minimal model.

The existence of good minimal models is a central goal of the minimal model program. It is known when $K_X$ is big, but the pseudo-effective non-big case remains open in general.