Generalized Prokhorov–Shokurov conjecture for generalized pairs

Let $(X',B'+M')$ be a generalized sub-pair with data $X\to X'$ and $M$, where $B'$, $M'$ and $M$ are $\mathbb{Q}$-divisors. Let $f\colon X'\to Z'$ be a contraction such that

$$K_{X'}+B'+M'\sim_{\mathbb{Q},f}0.$$

Assume that $M$ is semi-ample, let $c$ be the minimum positive integer such that $|cM|$ is basepoint-free, and assume that $(X',B'+M')$ is generalized klt over the generic point of $Z'$. Generalized Prokhorov–Shokurov conjecture. The moduli b-divisor $\mathbf{M}_{Z'}$ is b-semi-ample; for the generic fiber $X'_\eta$ of $f$, there is an integer $I_0$ depending only on $\dim X'_\eta$, the multiplicities of $B^h$, and $c$ such that

$$I_0(K_{X'_\eta}+B'_\eta+M'_\eta)\sim 0,$$

where the representative of $M'$ may be replaced within its $\mathbb{Q}$-linear equivalence class; and $\mathbf{M}_{Z'}$ is effectively b-semi-ample: there is a positive integer $I_1$ depending only on the dimension of $X'$, the horizontal multiplicities of $B$, and $c$ such that $I_1\mathbf{M}_{Z'}$ is very b-semi-ample, namely $I_1\mathbf{M}_{Z'}=\overline{L}$ for a basepoint-free divisor $L$ on some birational model of $Z'$. This is proposed as a generalization of the Prokhorov–Shokurov conjecture in the setting of generalized pairs; its status is not resolved in the supplied source context.