Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets. I have the following questions

1. When we have a quasi-isomorphism between $\Omega^{*}_{\text{poly}}\left(K\times L\right)$ and $\Omega^{*}_{\text{poly}}\left(K\right)\otimes \Omega^{*}_{\text{poly}}\left( L\right)$?

2. When we have an ISOMORPHISM?(Conjecture: when $K$ or $L$ is a finite simplicial set)

Here $dg_{\geq 0}Comm_{+}$ is the category of commutative unitary cochain differential graded algebras over a field of char zero.

Thanks!!!