Generalised Shokurov conjecture under bounded fibre volume

Let $d,v$ be natural numbers and let $\epsilon$ be a positive real number. Consider a generalised pair with data $X'\to X\overset{f}\to Z$ and $M'$, where $f$ is a contraction, and let $F$ be a general fibre of $f$. Assume that $(X,B+M)$ is generalised $\epsilon$-lc of dimension $d$, $K_X+B+M\sim_\mathbb R0/Z$, and there is an integral divisor $G\geq0$ on $X$ such that

$$0<\operatorname{vol}((B+M+G)|_F)<v.$$

Generalised Shokurov conjecture under bounded fibre volume. There is a positive real number $\delta$, depending only on $d,v,\epsilon$, such that the coefficients of the discriminant b-divisor $\mathbf{B}_Z$ lie in $(-\infty,1-\delta]$. This strengthens the preceding generalised conjecture by replacing the bigness of $-K_X$ over $Z$ with a weaker bounded-volume condition on the general fibres; it is posed as an open conjecture in the paper.