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?