Popa's conjecture on cohomological support loci and variation

Let $f\colon X\to A$ be a fibration from a smooth projective variety $X$ to an abelian variety $A$. For a coherent sheaf $\mathcal{F}$ on $A$, write

$$V^0(A,\mathcal{F})=\{\alpha\in\operatorname{Pic}^0(A)\mid H^0(A,\mathcal{F}\otimes\alpha)\neq0\}.$$

Let $\operatorname{Var}(f)$ denote the variation of the fibration $f$. Popa's conjecture. For every integer $m>1$ such that $f_*\omega_X^{\otimes m}\neq0$,

$$\dim V^0(A,f_*\omega_X^{\otimes m})\ge\operatorname{Var}(f).$$

In the case when $\kappa(X)=0$ and $f$ is the Albanese morphism of $X$, this conjecture is essentially equivalent to Ueno's Conjecture K, which predicts that, up to birational equivalence, $f$ becomes a projection onto a factor after an étale base change. The general conjecture is presented as a proposal by Mihnea Popa; no resolution is supplied in the source.