Relative Fujita-type freeness conjecture for mixed pluricanonical sheaves

Let $f:X\to Y$ be a surjective morphism between smooth projective varieties with $\dim Y=n$, let $\mathcal L$ be any ample invertible sheaf on $Y$, and let $k$ be a positive integer. Relative Fujita-type freeness conjecture. The sheaf

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

is generated by global sections for $l\geq n+1$. The paper presents this as a new conjecture similar to the Popa–Schnell conjecture and as a generalization of Fujita's freeness conjecture; its resolution is not given.