The following statement should be immediately implied by Eilenberg–Zilber theorem if the sequences $(i_0,\ldots,i_k)$ below are only monotone. But I need the strict monotone version which I believe to be true, as I have checked it for degree 0 and 1.
Let $A=(A_k)_{k\geqslant 0}$ be a cosimplicial object in an abelian category. Denote by $\partial_k^r:A_{k-1}\rightarrow A_k$ and $\sigma_k^r:A_{k+1}\rightarrow A_k$ its coface and codengeneracy maps respectively. We have the associated unnormalized cochain complex $$C^{*}(A):~ 0\xrightarrow{} A_0\xrightarrow{\delta_1} A_1\xrightarrow{\delta_2} A_2\xrightarrow{\delta_3} \cdots $$ where $\displaystyle \delta_k=\sum_{r=0}^k (-1)^r\partial_k^r$.
Now let $n$ be a positive integer. Denote by $\Delta_k^n$ the set of sequences $(i_0,\ldots,i_k)$ of integers satisfying $0\leqslant i_0<\cdots <i_k\leqslant n$. Consider the following cochain complex $$C^{*}_{\Delta^n}(A):~ 0\xrightarrow{} A_0^{\oplus\Delta^n_0}\xrightarrow{\delta'_1} A_1^{\oplus\Delta^n_1}\xrightarrow{\delta'_2} \cdots \xrightarrow{\delta'_n} A_n^{\oplus\Delta^n_n}\xrightarrow{} 0 $$ where for each element $\mathbf{a}=(a_{i_0,\ldots,i_k})\in A_k^{\oplus\Delta^n_k} $, $$\left(\delta'_{k+1}(\mathbf{a})\right)_{j_0,\ldots,j_{k+1}}:= \sum_{r=0}^{k+1}(-1)^r \partial_{k+1}^r(a_{j_0,\ldots,\hat{j_r},\ldots,j_{k+1}}).$$
Finally, let $Tr^{\leqslant n}C^*(A)$ be the truncation of $C^*(A)$ by forgetting the terms $C^k(A)$ for all $k>n$.
We have the obvious diagonal map $D:Tr^{\leqslant n}C^*(A)\rightarrow C^{*}_{\Delta^n}(A)$ which is a chain map.
Is $D$ a quasi-isomorphism?
