Smooth fibration conjecture for varieties with maximal PLI one-forms

Let $X$ be a smooth minimal model, and let $g$ be the maximal number of pointwise linearly independent (PLI) holomorphic one-forms that $X$ carries. Smooth fibration conjecture. There exists a smooth morphism from $X$ to a $g$-dimensional abelian variety. This conjecture proposes that the first conclusion of the paper's structure theorem remains valid without the Kodaira-dimension assumption; related results are known in low dimensions, but the general case remains open.