Generalized log minimal model and termination conjecture for lc pairs

Let $(X/Z,B+A)$ be a log canonical pair, where $B,A\ge 0$ are $\mathbb Q$-divisors, $A$ is $\mathbb Q$-Cartier, and the morphism $f\colon X\to Z$ is surjective. Assume

$$K_X+B+A\sim_{\mathbb Q}0/Z.$$

Generalized log minimal model conjecture. The pair $(X/Z,B)$ has either a Mori fibre space or a log minimal model $(Y/Z,B_Y)$; if $K_Y+B_Y$ is nef$/Z$, then it is semi-ample$/Z$; and if $(X/Z,B)$ is $\mathbb Q$-factorial dlt, then every LMMP$/Z$ on $K_X+B$ with scaling of an ample$/Z$ $\mathbb Q$-divisor terminates.

This statement is a general form of conjectures related to singularities and is motivated by the existence of log flips for log canonical pairs. The paper proves that it follows from either the semi-ampleness conjecture recorded separately below or the ACC conjecture on log canonical thresholds.