Let $X_\bullet := ... X_2 \to X_1$ be a tower of connected and simple spaces with the following properties:
The induced tower $H_\ast(X_\bullet; \mathbb{F}_p)$ of graded $\mathbb{F}_p$-vector spaces is Mittag-Leffler and lifts to a tower of graded abelian hopf algebras over $\mathbb{F}_p$. The induced tower $\pi_1(X_\bullet)$ is Mittag-Leffler.
By a theorem of Goerss the canonical morphism $$ H_\ast(holim X_\bullet; \mathbb{F}_p)\to lim H_\ast(X_\bullet; \mathbb{F}_p)$$ is an isomorphism, where the limit on the right hand side is taken in the category of graded abelian hopf algebras over $ \mathbb{F}_p$.
The limit in the category of graded abelian hopf algebras over $ \mathbb{F}_p$ forgets to the limit in the category of graded cocommutative coalgebras over $\mathbb{F}_p$ but does generally not forget to the limit in graded $\mathbb{F}_p$-vector spaces.
Is there a similar result on the chain level?
Is it true under the assumptions on $X_\bullet$ that the canonical morphism $$ C_\ast(holim X_\bullet; \mathbb{F}_p)\to holim C_\ast(X_\bullet; \mathbb{F}_p)$$ is a quasi-isomorphism, where $C_\ast(-; \mathbb{F}_p)$ are chains with $\mathbb{F}_p$-coefficients, and the homotopy limit on the right hand side is taken in the $\infty$-category of $E_\infty$-coalgebras over $\mathbb{F}_p$?
This would follow of course from Goerss theorem if homology $H_\ast$ would send the homotopy limit in $E_\infty$-coalgebras (over $\mathbb{F}_p$) of the tower $ C_\ast(X_\bullet; \mathbb{F}_p)$ to the limit in graded cocommutative coalgebras over $\mathbb{F}_p$.
Does one know such a result?
Can one say more if one additionally assumes that the tower $X_\bullet := ... X_2 \to X_1$ of spaces refines to a tower of grouplike $E_\infty$-spaces?
Under this assumption the induced tower $C_\ast(X_\bullet; \mathbb{F}_p)$ is a tower of $E_\infty$-hopf algebras over $\mathbb{F}_p$, i.e. abelian group objects (in the derived sense) in the $\infty$-category of $E_\infty$-coalgebras over $\mathbb{F}_p.$
Therefore by Goerss theorem the canonical morphism $$ C_\ast(holim X_\bullet; \mathbb{F}_p)\to holim C_\ast(X_\bullet; \mathbb{F}_p)$$ would be a quasi-isomorphism if homology $H_\ast$ would send the homotopy limit in $E_\infty$-hopf algebras of the tower $ C_\ast(X_\bullet; \mathbb{F}_p)$ to the limit in graded abelian hopf algebras over $\mathbb{F}_p$.
Does one know such a result?