Homotopy-limit conjecture for quotient stacks by reductive group schemes

Let $G$ be a reductive group scheme, let $X$ be a space with the quotient simplicial object $[X/G]_{\bullet}$, and let $Cpx$ denote the functor sending a space to the dg-category of complexes of sheaves of modules on it. The totalization construction associated with the group action is defined as in the preceding proposition. Homotopy-limit conjecture. The construction in Proposition still gives the homotopy limit of

$$Cpx([X/G]_{\bullet}).$$

For discrete $G$, the analogous assertion is proved when the functor sends finite coproducts to products; the conjecture addresses the non-discrete reductive case, where the associated cosimplicial diagram need not be Reedy fibrant. In particular, it predicts that totalization nevertheless computes the homotopy limit.