Effective semi-ampleness conjecture of the moduli part

Let $f\colon X\to Z$ be a klt-trivial fibration of relative dimension $k$, with $(X,B)$ a klt projective pair whose boundary $B$ has rational coefficients. Let $r$ be the positive integer associated with the fibration, and let $M_{Z'}$ denote the trace of the moduli b-divisor on a birational model $Z'\to Z$. Effective semi-ampleness conjecture of the moduli part. For every pair of positive integers $k,r$, there exists a positive integer $m=m(k,r)$ such that, for every such fibration, one can choose a birational model $Z'\to Z$ on which the moduli b-divisor descends and $mM_{Z'}$ is base point free.

This strengthens the expected semi-ampleness of the moduli part, which is known to be a nef rational divisor on a sufficiently high birational model in the stated klt rational-coefficient setting. The conjecture asserts a uniform effective multiple depending only on the relative dimension and the integer $r$.