The uniqueness conjecture for PD algebras with nonzero second cohomology

Let $V_1$ and $V_2$ be PD algebras satisfying the assumptions of Proposition~, except that their second cohomology is allowed to be nonzero, and suppose

$$\Hmlg^2(V_1)\simeq\Hmlg^2(V_2)\neq 0.$$

Here a PDGA quasi-isomorphism is a quasi-isomorphism preserving the Poincaré duality structure and orientation. Uniqueness conjecture. Proposition~ should remain valid: there should exist a PD algebra $V_3$, not required to be simply-connected, together with PDGA quasi-isomorphisms

$$V_1\longrightarrow V_3\longleftarrow V_2.$$

The source gives an example showing that simple connectedness of $V_3$ cannot in general be retained when the second cohomology is nonzero; the conjecture proposes that omitting this condition suffices.