Let $X$ be a scheme and $F$ be an injective object of $\mathrm{Qcoh}(X)$. Is it true that $F$ is acyclic with respect to the usual sheaf cohomology?
For noetherian schemes $X$ this is well-known; then $F$ even turns out to be flasque. I don't really care for pathological schemes, but I would like to know if it's true for quasi-compact quasi-separated schemes.
If $X$ is affine, then it is also well-known (no matter if $X$ is noetherian or not), because then actually every quasi-coherent module is acyclic. Remark that even on an affine scheme, whose underlying topological space is noetherian, there are injective quasi-coherent modules which are not flasque (SGA 6, Exp. II, App. I); but of course this does not influence the answer.
The background is that I would like to define cohomology within $\mathrm{Qcoh}(X)$, without using the category of (not necessarily quasi-coherent) $\mathcal{O}_X$-modules or even all sheaves on $X$. This works because $\mathrm{Qcoh}(X)$ is a Grothendieck category (without any assumptions on $X$), thus has enough injective objects. This cohomology would turn out to be the usual sheaf cohomology (i.e. computed in $\mathrm{Sh}(X)$) if and only if injective objects are acyclic with respect to the usual sheaf cohomology.
EDIT: a-fortiori answers the question affirmatively if $X$ is quasi-compact and semi-separated. Is there any chance to get the result also when $X$ is just assumed to be quasi-compact and quasi-separated? I've already convinced myself that the proof cannot be translated verbatim.
