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 the cohomology. 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 $B$.

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. However, this does provide certain uniqueness 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?