Decomposable Iitaka fibration conjecture without the Q-factorial assumption

Let $d$ be a positive integer and let $\Gamma\subset[0,1]$ be a DCC set. A $(\Gamma_0,\Gamma')$-decomposable Iitaka fibration is the decomposition notion defined for invariant Iitaka fibrations; its strong version additionally requires the relevant rounded linear systems to be birational.

Decomposable Iitaka fibration conjecture. There exist a positive integer $m$, a finite set $\Gamma_0\subset(0,1]$, and a DCC set $\Gamma'\subset[0,1]$, depending only on $d$ and $\Gamma$, such that every lc pair $(X,B)$ of dimension $d$ with $B\in\Gamma$, $\kappa_\iota(X,K_X+B)\ge0$, and either finite $\Gamma$ or all components of $B$ $\mathbb Q$-Cartier, has both the weak and strong decomposable Iitaka fibration properties.

This is presented as a stronger variant of the earlier decomposable Iitaka fibration conjecture and remains open in the supplied context.