By the work of Hamilton and Lazarev, cyclic $C_{\infty}$-algebras are uniquely determined up to cyclic $C_{\infty}$-quasi-isomorphism by the cyclic (PD) algebra structure on thetheir cohomology. It is a beautiful result. More precisely:
Suppose $A$ and $B$ are simply connected cyclic $C_{\infty}$-algebras which are $C_{\infty}$-quasi-isomorphic and the induced isomorphism in cohomology is one of PD algebras. Then there exists a cyclic $C_{\infty}$-quasi-isomorphism between $A$ and $B$. The statement does not hold for $A_{\infty}$ and $L_{\infty}$-algebras.
Actually, something a bit stronger is true. It is the following version of the transfer theorem:
Suppose A is a cyclic $C_{\infty}$-algebra. Then there exists a cyclic $C_{\infty}$-algebra structure on the cohomology $H(A)$ such that:
the pairing in $H(A)$ is induced by the pairing in $A$
the $C_{\infty}$-algebra structure on $H(A)$ in particular has structure maps $m_1=0$, $m_2=$ the product induced by the product of $A$.
there is a cyclic $C_{\infty}$-quasi-isomorphism between $A$ and $H(A)$
the cyclic $C_{\infty}$-algebra structure on $H(A)$ is unique up to cyclic $C_{\infty}$-ISOMORPHISM.
PD CDGA's are particular examples of cyclic $C_{\infty}$-algebras. Note that this does not give a positive answer to the question since having a cyclyc $C_{\infty}$-quasi-isomorphism between two PD CDGA's is weaker than having a zig zag with intermediate PD CDGAs. However, this does provide certain uniqueness for the PD models which, for example, is useful for studying the invariance of certain string topology operations at the level of cohomology.
Hamilton and Lazarev wrote several papers about this. They called cyclic $C_{\infty}$-algebras "symplectic" $C_{\infty}$-algebras. A relevant paper is "Symplectic $C_{\infty}$-algebras" which appeared in Moscow Mathematical Journal in 2008.
@Najib Idrissi, I wonder if this implies your result?