Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover.

Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such that $\mathcal{F}(\text{Spec}(A_i))$ has a finitely presented $A_i$-submodule $M_i\subset \mathcal{F}(\text{Spec}(A_i))$ for each $i$.

Does there exist a finitely presented $\mathcal{O}_X$-submodule $\mathcal{F}_0\subset\mathcal{F}$ such that $\mathcal{F}_0(\text{Spec}(A_i))$ contains $M_i$ as an $A_i$-submodule?

**Attempt** Since $X$ is separated, calling $j_k : \text{Spec}(A_k)\to X$ the open immersion, $j_k$ is quasi-compact, so $(j_k)_*(\widetilde{M}_k)$ is quasi-coherent. Take the sum and call $\mathcal{F}_0$ the image of the map $\bigoplus_{k=1}^n (j_k)_*(\widetilde{M}_k)\to\mathcal{F}$.
This attempt doesn't work, since for quasi-coherent sheaves the map $\text{id}\to j_*j^*$ is almost never an isomorphism, so there usually is no useful map $\bigoplus_{k=1}^n (j_k)_*(\widetilde{M}_k)\to\mathcal{F}$.
The question remains, however.

SloganI'd like to know if, for a quasi-coherent sheaf that on each member of a finite affine cover has a finitely presented sub-sheaf, has a finitely presented subsheaf whose restrictions on each member of that cover contain each of them.