Semi-ampleness and Kodaira-dimension conjecture for the moduli part of maximal-moduli fibrations

Let $(X/Z,B)$ be a GLC fibration, meaning that $(X,B)$ is a log pair and $f\colon X\to Z$ is a projective surjective morphism with connected fibres between normal quasi-projective varieties, with $(X,B)$ log canonical above the generic point of $Z$. Assume that $(X,B)$ is log canonical, $B\geq 0$, $(X/Z,B)$ has maximal moduli, $M_X$ is $f$-nef, and $M_X$ is BP stable. Here $M_X$ denotes the moduli part of the fibration, and $\kappa(X/Z,B)$ and ${\rm var}(X/Z,B)$ denote the relative Kodaira dimension and variation, respectively.

Variant of Shokurov's b-semi-ampleness conjecture. For all $m\gg 0$ and any $\epsilon\in\mathbb Q_{>0}$, the divisor

$$M_X+\epsilon f^*\operatorname{det} f_*\mathcal O(mM_X)$$

is semi-ample; moreover,

$$\kappa(M_X)\geq\kappa(X/Z,B)+{\rm var}(X/Z,B).$$

The statement is a variant of Shokurov's conjecture on semi-ampleness of the moduli part for generically log canonical fibrations. The preceding discussion explains that ordinary semi-ampleness can fail for general GLC fibrations, while the conjectured perturbation by a divisor pulled back from the moduli space of fibres is intended to address this obstruction. The source provides no resolution status for this variant.