Toën's full-faithfulness conjecture for the functor from derived schemes to geometric stacks

Let $k$ be the base ring, let $DG$-$Sch$ denote the homotopy category of derived schemes, and let $C(k)_{\mathbb{U}}$-$Aff^{\sim,\operatorname{ffqc}}$ denote the homotopy theory of geometric stacks over the model category of complexes. The construction in the preceding proposition gives a functor

$$\Theta: Ho(DG\text{-}Sch) \longrightarrow Ho(C(k)_{\mathbb{U}}\text{-}Aff^{\sim,\operatorname{ffqc}}).$$

Full-faithfulness conjecture. The functor $\Theta$ is fully faithful. This would embed the homotopy theory of $DG$-schemes into the homotopy theory of geometric stacks over the model category of complexes. The supplied text gives no evidence of a resolution, so the conjecture remains open.