Conjecture \[z-conj1.2\] in [@denf], [@dutta], [@dutta-murayama], and [@iwai]. In this paper, we do not directly treat Conjecture \[z-conj1.2\]. We propose a new conjecture similar to Conjecture \[z-conj1.2\]. It is also a generalization of Conjecture \[z-conj1.1\].

\[z-conj1.3\] 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 $Y$. Then, for every positive integer $k$, the sheaf $$f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal L^{\otimes l}$$ is generated by global sections for $l\geq n+1$.

If Conjecture \[z-conj1.3\] and Fujita’s original freeness conjecture (see Conjecture \[z-conj1.1\]) hold true, then $f_*\omega^{\otimes k}_X\otimes \mathcal 
L^{\otimes l}$ is generated by global sections for every $l\geq k(n+1)$ since $$f_*\omega^{\otimes k}_X\otimes \mathcal L^{\otimes l}\simeq f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal L^{\otimes \left(l-(k-1)(n+1)\right)}\otimes 
\left(\omega_Y\otimes 
\mathcal L^{\otimes n+1}\right)^{\otimes k-1}.$$ Therefore, Conjecture \[z-conj1.3\] is sharper than Conjecture \[z-conj1.2\].

It is well known that Conjecture \[z-conj1.3\] holds true when $Y$ is a curve. This means that $f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes 
\mathcal L^{\otimes l}$ is generated by global sections for every $l\geq 2$. More generally, we have:

\[z-thm1.4\] Let $f:X\to C$ be a surjective morphism from a smooth projective variety $X$ onto a smooth projective curve $C$. Let $\mathcal H$ be an ample invertible sheaf 