MRC-fibration conjecture for pseudoeffective differential forms

Let $X$ be a uniruled projective manifold, let $f:X\dasharrow Z$ be its MRC fibration, and assume that $Z$ is a projective manifold. Set $n=\dim X$, and let $1\leq q\leq n$. The sheaves of holomorphic differential forms in degree $q$ on $X$ and $Z$ are denoted by $\Omega^q_X$ and $\Omega^q_Z$, respectively.

MRC-fibration pseudoeffectivity conjecture. One has

$$\Omega^q_X\text{ is pseudoeffective}\quad\Longleftrightarrow\quad\Omega^q_Z\text{ is pseudoeffective}.$$

This proposes that, for a uniruled manifold, pseudoeffectivity of differential forms is detected on the base of the maximal rationally connected fibration. The preceding results establish related implications and the converse in certain degree ranges, but the full equivalence remains open in the supplied text.