Decomposable Iitaka fibration conjecture

Let $d$ be a positive integer and let $\Gamma\subset[0,1]$ be a DCC set. A $(\Gamma_0,\Gamma')$-decomposable Iitaka fibration means that $B=\sum_i a_iB_i$, where $\Gamma_0=\{a_1,\ldots,a_k\}\subset(0,1]$ is finite with $\sum_i a_i=1$, each $B_i$ has coefficients in $\Gamma'$, each $(X,B_i)$ is lc, and every invariant Iitaka fibration of $K_X+B$ is an Iitaka fibration of $K_X+B_i$. The $(m,\Gamma_0,\Gamma')$-version additionally requires $|\lfloor m(K_X+B_i)\rfloor|$ to be birational to every invariant Iitaka fibration of $K_X+B$.

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 $\mathbb Q$-factorial lc pair $(X,B)$ of dimension $d$ with $B\in\Gamma$ and $\kappa_\iota(X,K_X+B)\ge0$ has both a $(\Gamma_0,\Gamma')$-decomposable Iitaka fibration (weak version) and an $(m,\Gamma_0,\Gamma')$-decomposable Iitaka fibration (strong version).

This conjecture proposes a uniform perturbation of an invariant Iitaka fibration into an effective log Iitaka fibration and remains open.