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I need either a proof or a good reference for the following plausible statement:

Let $S$ be a scheme and let $C$ be a bounded complex of abelian sheaves on $S_{\rm{fppf}}$. Let $S^{\prime}\rightarrow S$ be a finite etale Galois cover of $S$ with Galois group $G$. Then there exists a hypercohomology Hochschild-Serre spectral sequence in flat (fppf) cohomology $$ H^{r}(G,{\mathbb H}^{s}(S',C))\implies {\mathbb H}^{r+s}(S,C). $$ This spectral sequence has been used by B.Kahn and E.Peyre when $S$ is the spectrum of a field. When C is a 1-term complex (i.e., simply an abelian sheaf on $S_{\rm{fppf}}$), the sequence is explicitly stated in Milne's Etale Cohomology book (bottom of p.105). The only question is whether the hypercohomology case has been handled by someone somewhere. Thanks in advance!

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