Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where $\tau$ is a Grothendieck topology on $C$. We have a canonical functor $$\mathbb{Z}_{\tau}(-):C\rightarrow Ab(C_{\tau}).$$ Let $ch\big(Ab(C_{\tau})\big)$ be the category of unbounded chain complexes on $Ab(C_{\tau})$, and let $D\big(Ab(C_{\tau})\big)$ be its derived category. Since the $qfh$-topology is finer than the Nisnevich topology, we have an adjunction $Ab(C_{Nis})\rightleftarrows Ab(C_{qfh})$.
Let $X$ be any $k$-scheme in $C$.
- QUESTION
Is the canonical morphism $$\mathbb{Z}_{Nis}(X)\rightarrow \mathbb{Z}_{qfh}(X)$$ viewed as a chain complex concentrated in degree zero, an isomorphism in the derived category $D\big(Ab(C_{Nis})\big)\otimes\mathbb{Q}$ with rational coefficients?
Thanks in advance.