4 removed a reference to an incorrect solution

Ok, this question is much less ambitious than it might sound, but still:

Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-isomorphisms. There is a similar definition for not necessarily commutative differential graded algebras (dga's).

1. If two $\mathbf{Q}$-cdga's, $A$ and $B$, are quasi-isomorphic as dga's, are they necessarily quasi-isomorphic as cdga's? I suspect that the answer is no, but don't know any counter-examples, nor can prove that such counter-examples exist.

2. Same question as 1 when $A$ and $B$ are the Sullivan $\mathbf{Q}$-polynomial cochain algebras of simply connected compact polyhedra. In other words, is the "rational noncommutative homotopy type" of compact simply-connected polyhedra the same as the usual rational homotopy type?

upd: Contrarily to what I had thought, the answer to 1. (and hence, to 2.) appears to be yes. Thanks to Boris, James, Joey and Tyler!

3 update

Ok, this question is much less ambitious than it might sound, but still:

Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-isomorphisms. There is a similar definition for not necessarily commutative differential graded algebras (dga's).

1. If two $\mathbf{Q}$-cdga's, $A$ and $B$, are quasi-isomorphic as dga's, are they necessarily quasi-isomorphic as cdga's? I suspect that the answer is no, but don't know any counter-examples, nor can prove that such counter-examples exist.

2. Same question as 1 when $A$ and $B$ are the Sullivan $\mathbf{Q}$-polynomial cochain algebras of simply connected compact polyhedra. In other words, is the "rational noncommutative homotopy type" of compact simply-connected polyhedra the same as the usual rational homotopy type?

upd: Contrarily to what I had thought, the answer to 1. (and hence, to 2.) appears to be yes. Thanks to Boris, James, Joey and Tyler!

2 rephrased the title

Ok, this question is much less ambitious than it might seemsound, but still:

Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-isomorphisms. There is a similar definition for not necessarily commutative differential graded algebras (dga's).

1. If two $\mathbf{Q}$-cdga's, $A$ and $B$, are quasi-isomorphic as dga's, are they necessarily quasi-isomorphic as cdga's? I suspect that the answer is no, but don't know any counter-examples, nor can prove that such counter-examples exist.

2. Same question as 1 when $A$ and $B$ are the Sullivan $\mathbf{Q}$-polynomial cochain algebras of simply connected compact polyhedra. In other words, is the "rational noncommutative homotopy type" of compact simply-connected polyhedra the same as the usual rational homotopy type?

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