Fujino's relative Fujita-type conjecture for direct images of relative pluricanonical bundles

Let $f:X\to Y$ be a surjective morphism of smooth projective varieties over an algebraically closed field of characteristic zero, with $Y$ of dimension $n$, and let $\mathcal L$ be an ample line bundle on $Y$. For an integer $m\geq 1$, consider the sheaf $f_*\omega_{X/Y}^m\otimes\omega_Y$. Fujino's conjecture. For each $m\geq 1$, the sheaf

$$f_*\omega_{X/Y}^m\otimes\omega_Y\otimes\mathcal L^l$$

is generated by its global sections for $l\geq n+1$. This is a generalization of Fujita's freeness conjecture. The source presents it as a proposed conjecture and does not give evidence that it is resolved in the stated generality.