The Mori MMP termination and Mori fibre space conjecture

Let $(Z,\Phi)$ be a Kawamata log terminal pair. Run the $(K_Z+\Phi)$-minimal model program via a birational map $f\colon Z\dashrightarrow X$, and write $\Delta=f_*\Phi$. Mori MMP conjecture. The program can be run so that either $(X,\Delta)$ is a log terminal model, meaning that $K_X+\Delta$ is nef, or $X$ admits a Mori fibre space $\phi\colon X\to S$ with $\rho(X/S)=1$ and $-(K_X+\Delta)$ $\phi$-ample. The conjecture concerns the general case of the minimal model program; the cited results establish finiteness of log terminal models when either $\Phi$ or $K_Z+\Phi$ is big, while the remaining pseudo-effective case is described as unknown.