Numerically flat or positively twisted subsheaf conjecture for abelian fibrations

Let $X$ be a projective manifold such that the cotangent bundle $\Omega_X$ is pseudoeffective. Assume that $X$ admits an abelian fibration $f:X\to C$ onto a curve $C$, and let $F$ be a general fibre of $f$. Then the subsheaf $\mathcal V\subset\Omega_X$ contains a non-zero reflexive subsheaf $\mathcal W$ such that one of the following holds: $\mathcal W$ is numerically projectively flat, or, for some $\varepsilon>0$, the $\mathbb Q$-twist $\mathcal W\otimes\mathcal O_X(-\varepsilon F)$ is pseudoeffective.