Let $X$ be a connected scheme, smooth and proper over $\mathbb{C}$. Let $F$ be a locally free $\mathcal{O}_X$-module of finite rank $r>1$. Suppose on a non-empty affine open $U\subset X$ whose complement is irreducible we have an isomorphism of $\mathcal{O}_X$-modules $f:\mathcal{O}^{\oplus r}_X|_{U}\rightarrow F|_{U}$.

Does there necessarily exist a locally free $\mathcal{O}_X$-module $L$ of rank $1$ with an identification $\mathcal{O}_X|_U\rightarrow L|_U$ such that there is a morphism of $\mathcal{O}_X$-modules $g:L^{\oplus r}\rightarrow F$ with $g|_{U}=f$?

EDIT: I believe that if $D=X/U$ (which is a Weil divisor because $X$ is separated and thus a Cartier divisor because $X$ is smooth) has positive Iitaka dimension, then this is true. To find $g$, we need $\mathcal{O}_X$-independent morphisms $L\rightarrow F$, which are in bijection with global sections of $L^\vee \otimes_{\mathcal{O}_X}F$. So if we take $L=\mathcal{O}(-mD)$ for sufficiently large $m$, then $L^\vee \otimes_{\mathcal{O}_X}F$ is going to have $r$ independent global sections and we win. But does every smooth proper $\mathbb{C}$-scheme have a divisor of positive Iitaka dimension?