Relative semi-ampleness conjecture for dlt pairs

Let $(X/Z,B)$ be a $\mathbb Q$-factorial dlt pair, and let

$$T:=\left\lfloor B\right\rfloor.$$

Assume that $B$ is a $\mathbb Q$-divisor and $Z$ is affine. Suppose that $K_X+B$ is nef$/Z$, that $(K_X+B)|_S$ is semi-ample$/Z$ for every component $S$ of $T$, and that $K_X+B-\epsilon P$ is semi-ample$/Z$ for some $\mathbb Q$-divisor $P\ge 0$ with $\operatorname{Supp} P=T$ and every sufficiently small rational number $\epsilon>0$. Semi-ampleness conjecture. Then $K_X+B$ is semi-ample$/Z$.

This is a relative form of a semi-ampleness statement used to derive the generalized log minimal model conjecture. The conjecture is known when $Z$ is a point, while the relative case remains open.