Homotopy commutativity of the homotopy shuffle product

Let $A^\bullet$ be a strict simplicial graded commutative DGA indexed by $I$, and let

$$B_{red}(\check{C}(I,A^\bullet))$$

be the reduced bar construction equipped with the homotopy shuffle product.

Homotopy shuffle commutativity conjecture. The homotopy shuffle product is homotopy commutative for every such $A^\bullet$. Consequently, the induced product

$$H^\bullet(\mu):H^\bullet(B_{red}(\check{C}(I,A^\bullet)))\otimes H^\bullet(B_{red}(\check{C}(I,A^\bullet))) \to H^\bullet(B_{red}(\check{C}(I,A^\bullet)))$$

is graded commutative.

The author states that neither homotopy commutativity of the product nor graded commutativity of the induced cohomology product is known in this generality. In characteristic zero, a Thom–Whitney construction provides a commutative DGA associated with a strict simplicial graded commutative DGA.