Given a fiber square of simplicial sets

$$\begin{array}{cc} & \hspace{-7mm} E \\ &\hspace{-7mm}\downarrow \\ \ast\longrightarrow &\hspace{-7mm} B \end{array}$$

and a homology theory $h(-)$, there is an associated Eilenberg-Moore spectral sequence converging to the homology of the homotopy fiber $F$. This is just the spectral sequence for a cosimplicial space, specifically a cobar construction $C^\bullet(E,B,\ast)$. It is claimed in this paper of Hopkins and Ravenel, in the middle of page 7, that if this Eilenberg-Moore spectral sequence converges strongly then $\Sigma_+^\infty Tot(C^\bullet(E,B,\ast))\simeq Tot(C^\bullet(\Sigma^\infty_+E,\Sigma^\infty_+ B,\mathbb{S}))$.

Why is this true? I know from an old paper of Bousfield that the EMSS for stable homotopy converges strongly in certain nice cases (e.g. if $B$ is simply connected), but does that give me an actual equivalence of objects? That only seems to tell me nice things about the algebra. For instance that there exists a finite filtration of $\pi_n^S(F)$ whose filtration quotients are the $E_{i,j}^\infty$ for $i+j=n$. That seems several steps away from making a general statement about the suspension spectrum functor commuting with totalization.

More generally, are there other functors $Top\to Spectra$ which commute in this fashion? For instance, it seems like the Thom spectrum functor (a sort of twisted suspension spectrum functor) for suitable fibrations should also commute with totalization is suspension spectrum does.

Given a fiber square of simplicial sets

$$\begin{array}{cc} & \hspace{-7mm} E \\ &\hspace{-7mm}\downarrow \\ \ast\longrightarrow &\hspace{-7mm} B \end{array}$$

and a homology theory $h(-)$, there is an associated Eilenberg-Moore spectral sequence converging to the homology of the homotopy fiber $F$. This is just the spectral sequence for a cosimplicial space, specifically a cobar construction $C^\bullet(E,B,\ast)$. It is claimed in this paper of Hopkins and Ravenel, in the middle of page 7, that if this Eilenberg-Moore spectral sequence converges strongly then $\Sigma_+^\infty Tot(C^\bullet(E,B,\ast))\simeq Tot(C^\bullet(\Sigma^\infty_+E,\Sigma^\infty_+ B,\mathbb{S}))$.

Why is this true? I know from an old paper of Bousfield that the EMSS for stable homotopy converges strongly in certain nice cases (e.g. if $B$ is simply connected), but does that give me an actual equivalence of objects? That only seems to tell me nice things about the algebra. For instance that there exists a finite filtration of $\pi_n^S(F)$ whose filtration quotients are the $E_{i,j}^\infty$ for $i+j=n$. That seems several steps away from making a general statement about the suspension spectrum functor commuting with totalization.

More generally, are there other functors $Top\to Spectra$ which commute in this fashion? For instance, it seems like the Thom spectrum functor (a sort of twisted suspension spectrum functor) for suitable fibrations should also commute with totalization is suspension spectrum does.

EDIT----------------------------

I've also discovered this mysterious correspondence between Tom Goodwillie and Mike Hopkins (this is how people talked about this stuff before MO I guess!) that shows that suspension commutes with Tot in certain cases.

http://www.lehigh.edu/~dmd1/tg7192