Birational abelian-variety fibration conjecture for varieties with nowhere-vanishing one-forms

Let $X$ be a smooth projective variety admitting a holomorphic one-form without zeros. Birational abelian-fibration conjecture. Then $X$ is birational to a smooth projective variety $X'$ admitting a smooth morphism

$$X'\to A$$

to a positive-dimensional abelian variety $A$. This would provide a geometric structure behind varieties carrying nowhere-vanishing holomorphic one-forms and, together with additivity of Kodaira dimension for analytic fibre bundles, would generalize the result of Popa and Schnell. The conjecture is stated without a resolution in the source.