For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices $$ \Phi : A \to B \otimes \Omega^*(\Delta^n) $$ and simplices maps induced by the inclusion and degeneracy maps on the standard simplices. Now forgetting the algebra structure of $A$ and $B$, we can also look at the mapping space in the category of cochain complexes $\text{Map}_{\mathsf{Ch}^*(\mathbb{Q})}(A,B)$ where the $n$-simplices are given by $$ \Phi \colon A \to B \otimes C^*(\Delta^n). $$
For each $n$, integration defines a cochain map $$ \int : \Omega^*(\Delta^n) \to C^*(\Delta^n) $$ into the simplicial cochains on the standard $n$-simplex. I'm fairly sure this commutes with the maps induced by inclusions and degeneracy maps, meaning we get a map of simplicial sets $$ \int_* : \text{Map}_{CDGA}(A,B) \to \text{Map}_{\mathsf{Ch}^*(\mathbb{Q})}(A,B) $$ where a map $\Phi : A \to B \otimes \Omega^*(\Delta^n)$ is sent to the composition $$ A \xrightarrow{\Phi} B \otimes \Omega^*(\Delta^n) \xrightarrow{1_B \otimes \int} B \otimes C^*(\Delta^n) $$
My question is: is the map $\int_*$ a Kan fibration?
So far I've really just been focusing on the case of finding lifts of the form $$\require{AMScd} \begin{CD} \Lambda^1[2] @>{}>> \text{Map}_{CDGA}(A,B);\\ @VVV @VV{\int_*}V \\ \Delta[2] @>{}>> \text{Map}_{\mathsf{Ch}^*(\mathbb{Q}}(A,B); \end{CD} $$ From these kinds of examples, I think that at minimum $A$ needs to be a Sullivan algebra. My reasoning is that $\text{Map}_{\mathsf{Ch}^*(\mathbb{Q})}(A,B)$ is always a Kan complex, so every map from $\Lambda^1[2] \to \text{Map}_{\mathsf{Ch}^*(\mathbb{Q})}(A,B)$ can be extended irrespective of whether $A$ is a Sullivan algebra or not. So, in this case, we should be able to find examples where the extension exists in $\text{Map}_{\mathsf{Ch}^*(\mathbb{Q})}(A,B)$ but not in $\text{Map}_{CDGA}(A,B)$.
Another observation is that $\int_*$ is not surjective (even on the $0$-simplices), since not every cochain map is an algebra map. I'm not sure if this is a problem or not either.
