Building on the work of Anderson and Rector, Bousfield's paper "On the homology spectral sequence of a cosimplicial space" constructs a spectral sequence which takes in a cosimplicial space (here space = simplicial set) $X$ and (under certain conditions) converges to the homology of $Tot(X)$, known as the generalized Eilenberg-Moore spectral sequence (GEMSS). Sticking with with integer coefficients, its $E^2$ page is given by $E^2_{s, t} \cong \pi^sH_tX$, i.e. the cohomology of the cosimplicial abelian group $H_tX$.
On the other hand, there is an adjunction between simplicial sets and simplicial abelian groups, with left adjoint the free abelian group functor $\tilde{\mathbb{Z}}$ and right adjoint the forgetful functor $U$. Note that for a simplicial set $Y$, we have $\pi_tU\tilde{\mathbb{Z}}Y \cong H_t(Y)$. Now consider the cosimplicial space $U\tilde{\mathbb{Z}}X$. The Bousfield-Kan spectral sequence (BKSS) associated to $U\mathbb{Z}X$ has $E^2$ page $E^2_{s, t} \cong \pi^s\pi_tU\tilde{\mathbb{Z}}X \cong \pi^sH_t(X)$.
In other words, the $E^2$ page of the GEMSS of $X$ is isomorphic, objectwise, to the $E^2$ page of the BKSS of $U\tilde{\mathbb{Z}}X$. My qusetion is this: Are these in fact the same spectral sequence? Is there a map of spectral sequences which induces this isomorphism on $E^2$ pages?
Some possible evidence that this might be true: Dwyer's ``Strong convergence of the Eilenberg-Moore spectral sequence'' constructs the EMSS in the following way: Given a fibration $F \to E \to B$, one considers the cobar construction $C(E, B, \ast$), which he calls F. He then constructs the EMSS as the BKSS (or homotopy spectral sequence) associated to $U\tilde{\mathbb{Z}}C(E, B, \ast) = U\tilde{\mathbb{Z}}$F. I suspect also this question is related to Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum? but I was not able to draw a firm conclusion from that earlier post.
