Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^*(\Delta^\bullet)\otimes A^\bullet$, where $C^*(-)$ is the simplicial cochain complex, i.e. $|A^\bullet|$ is the cochain complex with $|A^\bullet|^n = A^n$ and differential the alternating sum of coface maps:

- The Alexander-Whitney map endows the realization functor with a lax monoidal structure, so that it sends the monoid $A^\bullet$ in cosimplicial groups to a monoid in cochain complexes, i.e. a differential graded algebra. Explicitly, the product of $a_p\in A^p$ and $a_q\in A^q$ is (up to sign) $f_{p,q}(a_p)b_{p,q}(a_q)$, where $f_{p,q}: A^p\to A^{p+q}$ and $b_{p,q}:A^q\to A^{p+q}$ are the "front and back" cofaces. Note that this does not use the commutativity of $A^\bullet$, and the product is not commutative in general. Call this (dga, and thus in particular) $A_\infty$-algebra $|A^\bullet|_{CG}$.
- The coend $\Omega^*(A^\bullet):=\int^\Delta \Omega_P^*(\Delta^\bullet)\otimes A^\bullet$, where $\Omega^*_P(\Delta^n) = k[t_0,\dots,t_n,\mathrm dt_0,\dots,\mathrm dt_n]/(t_0+\dots+t_n-1,\mathrm dt_0 + \dots + \mathrm dt_n)$ carries a natural (commutative) dga structure. As explained by Cheng and Getzler, there is a simplicial retraction $\Omega_P^*(\Delta^\bullet) \rightleftarrows C^*(\Delta^\bullet)$, giving rise to a retraction $\Omega^*(A^\bullet)\rightleftarrows|A^\bullet|$ along which this structure can be transfered to an $A_\infty$-structure on $|A_\bullet|$ (actually, even a $C_\infty$-structure). The operations are given by sums over trees. Call this $A_\infty$-algebra $|A^\bullet|_{CG}$.

Obviously these two structures are quite different: For instance, the $2$-ary operation of the first is associative, but not commutative, whereas that of the second is commutative, but not associate.

There is a special case where I know that these $A_\infty$ algebras are equivalent: If $M$ is a smooth manifold and $A^\bullet = \operatorname{Sing}^\bullet(M)$ is the cosimplicial commutative algebra of functions on smooth simplices, there is a zigzag $$ |A^\bullet|_{CG}\rightarrow \Omega^*(A^\bullet)\rightarrow \int^\Delta \Omega^*(\Delta^\bullet)\otimes A^\bullet \leftarrow \Omega^*(M)\rightarrow |A^\bullet|_{AW} $$ where the maps are, in order:

- the canonical $A_\infty$-morphism produced by the homotopy transfer
- the (simplex-wise) inclusion of polynomial forms into smooth forms
- the map $\omega\mapsto (\sigma\mapsto \sigma^*\omega)$
- an explicit $A_\infty$-isomorphism obtained by Chen's iterated integrals, see for instance here.

Is there a natural $A_\infty$-isomorphism between $|A^\bullet|_{AW}$ and $|A^\bullet|_{CG}$? Can it be made reasonably explicit? If there is, does it extend the above construction for $A^\bullet = \operatorname{Sing}^\bullet(M)$, or is there at least a natural (and explicit?) $A_\infty$-homotopy between them?