Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let
$\mathrm{Cotor}_A(M,N)$ denote the cotorsion product of $M$ and $N$ relative to $A$.

The graded $K$-vector space $\mathrm{Cotor}_A(M,N)$ is by definition the homology of the totalization of the cosimplicial cochain complex over $K$ with $n$-th term $M \otimes A^{\otimes n} \otimes N$,
where the tensor product is in cochain complexes over $K.$

Let $X \to Y$ be a Serre fibration between connected spaces and $F$ its fiber over a given point $y$ of $Y.$

If $Y$ is simply connected, by a theorem of Eilenberg and Moore there is a canonical isomorphism
\begin{equation}
H_*(F;K)\cong \mathrm{Cotor}_{C_*(Y; K)}(C_*(X; K),C_*(*; K)), \ \ \ \ (**)
\end{equation}
where $C_*(-;K)$ are singular chains with coefficients in the field $K.$

Can we replace the condition that $Y$ is simply connected by a weaker condition?

Precisely, is there still a canonical isomorphism $(**)$ if $Y$ is simple and $\pi_1(Y)$ is a 
$K$-vector space?