Popa–Schnell's conjecture on direct images of pluricanonical bundles

Let the base field be an algebraically closed field of characteristic zero. Let $f:X\to Y$ be a morphism from a smooth projective variety $X$ to a smooth projective variety $Y$ of dimension $n$, and let $\mathcal L$ be an ample line bundle on $Y$. Popa–Schnell's conjecture. For every $m\ge 1$, the sheaf

$$f_*\omega_X^m\otimes\mathcal L^l$$

is generated by its global sections for $l\ge m(n+1)$. This conjecture is a relative form of Fujita's freeness conjecture for direct images of pluricanonical bundles; the paper generalizes the corresponding theorem to ample line bundles that are not globally generated and also studies positive characteristic.