Let $\pi:\mathcal{X} \to S$ be a flat, family of projective varieties (here $\mathcal{X}$ and $S$ are noetherian). Let $E$ and $F$ be two locally free sheaves on $\mathcal{X}$ such that for all $s \in S$, $E_s \cong F_s$, where $E_s$ and $F_s$ are the restriction of $E$ and $F$ respectively to the fiber $\mathcal{X}_s$ over $s$ of $\pi$. Then, does there exists an invertible sheaf $L$ on $S$ such that $E \cong F \otimes \pi^*L$?