The singular cochain complex of a space $X$ with coefficients in a ring $R$ can be endowed with a product, turning it into a differential graded $R$-algebra, such that in cohomology it gives the familiar cup product. However, at the chain level one has to be careful, since different choices of products can have different properties: e.g., an arbitrary one will not necessarily be associative, whereas the one induced by the Alexander–Whitney map will be. This indeterminacy is parametrized by the Eilenberg–Zilber theorem.
Now, consider an H-space $X$. Then in homology (with whichever coefficients) we have a product, the Pontrjagin product, which is associative or commutative if the product of $X$ is so too, up to homotopy. But what about at the chain level? The product at the chain level is given by $$C_*(X)\otimes C_*(X)\to C_*(X\times X)\to C_*(X),$$ where the first map is a fixed chain homotopy equivalence, and the second one is induced by the multiplication.
My question is:
If we take the first map to be the shuffle map and $X$ to be homotopy associative, is the resulting differential graded algebra $C_*(X)$ associative?
I'm guessing the analogous question for commutativity has a negative answer, since it doesn't work for cohomology; any comments on this are also welcome.