Does every sheaf embed into a quasicoherent sheaf?

Question

Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?

Remark. The coherator yields a way to find a quasicoherent sheaf together with a morphism to $\mathcal{E}$. But I'm interested in finding a quasicoherent sheaf together with a monomorphism from $\mathcal{E}$.

Motivation. There is a way to set up the theory of sheaf cohomology for quasicoherent sheaves without injective or flabby resolutions. If any sheaf of modules would embed into a quasicoherent one, we might be able to extend this development to arbitrary (not necessarily quasicoherent) sheaves of modules.

Answer 1

That already fails for $X$ equal to $\text{Spec}\ R$, where $R$ is a DVR with generic point $\eta = \text{Spec}\ K$. Since there are only two nonempty open subsets of $X$, namely all of $X$ and $\{\eta\}$, there is a straightforward equivalence between the category of $\mathcal{O}_X$-modules and the category of triples $(M,V,\phi)$ of an $R$-module $M$, a $K$-module $V$, and an $R$-module homomorphism $$\phi:M\to V.$$ This is quasi-coherent if and only if $\phi$ induces an isomorphism $$M\otimes_R K \xrightarrow{\cong} V,$$ i.e., the $\mathcal{O}_X$-module is equivalent to $$(M,M\otimes_R K,\iota_M).$$ In particular, consider the $\mathcal{O}_X$-module $$(R,\{0\},0).$$ For every $\mathcal{O}_X$-module homomorphism of this $\mathcal{O}_X$-module to a quasi-coherent $\mathcal{O}_X$-module, $$(\psi_R,\psi_\eta):(R,\{0\},0) \to (M,M\otimes_R K,\iota_M),$$ the composite $\iota_M\circ \psi_R$ equals $0$. Thus, the image $\psi_R(R)$ is contained in the torsion submodule of $M$. Every torsion quotient of $R$ has nonzero kernel. Thus, $(\psi_R,\psi_\eta)$ is not a monomorphism.