Let X and Y be to varieties and $F\colon D\mathrm{QCoh}(X) \to D\mathrm{QCoh}(Y)$ a continuous functor between the corresponding unbounded derived categories of quasi-coherent sheaves (given by a kernel on X×Y). Assume that $F(D^b\mathrm{Coh}(X)) \subseteq D^b\mathrm{Coh}(Y)$ and that F is conservative.

Are there any conditions that ensure that $F(\mathcal{G}) \in D^b\mathrm{Coh}(Y)$ implies $\mathcal{G} \in D^b\mathrm{Coh}(X)$?

The corresponding statement for the abelian categories is easy (if $\mathcal G$ is not coherent, then there is an infinite increasing sequence of coherent subsheaves giving rise to such a sequence of subsheaves of $F(\mathcal G)$, which has to stabilize), but I don't seem to be able to transport this proof to the derived setting.