Let $(X, O_X)$ be an arbitrary scheme and $\mathcal{F}$ a presheaf of $O_X$-modules. The presheaf $\mathcal{F}$ is said to be separated if the natural map to it's sheafification $\mathcal{F}^+$ is an injection. That is, for each open set $U$, $\mathcal{F}(U) \hookrightarrow \mathcal{F}^+(U)$.

Let $\mathcal{F}, \mathcal{G}$ be two sheaves of $O_X$ modules. Is the presheaf tensor product $\mathcal{F} \otimes_{O_X} \mathcal{G}$ a separated presheaf? Is this true more generally if $(X,O_X)$ is just a ringed space?

This tells you for example, that if $\mathcal{F}, \mathcal{G}$ have some global sections then so does their sheaf tensor product (assuming the presheaf tensor product has global sections).

I wasn't able to find this in the stacks project. Is this proved in books on sheaf theory?