In the paper by Bhatt and Scholze on prismatic cohomology (https://arxiv.org/pdf/1905.08229.pdf), it is stated that the de Rham comparison theorem for prismatic cohomology can be lifted to an equivalence of cdga's. This confuses me, because I don't see why the de Rham complex of a smooth and proper scheme over a field k of characteristic p which is not affine is a cdga. It seem to me that it might be an $E_\infty$ algebra which is not strictly commutative. So i'm asking:
It it true that for a smooth and proper scheme X over a field k of characteristic $p$, the global sections of the de Rham complex $R\Gamma(X,DR_X)$, which is a priori an $E_\infty$ algebra in the $\infty$-category of $k$-modules, arises from a commutative differential graded algebra $k$-vector spaces? Note that this imposes many restrictions on the complex, namely the vanishing of most of Steenrod reduced powers on its cohomology.
If it is a cdga, why is it the case?, or, alternatively, if it is not the case, what do I miss in the prismatic version of de Rham comparison stated in this paper?