TITLE: Noncommutative rational homotopy type
QUESTION [32 upvotes]: 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).

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.
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?

REPLY [11 votes]: An affirmative answer (to both questions) appeared today on the arxiv, due to Campos, Petersen, Robert-Nicoud, Wierstra: https://arxiv.org/abs/1904.03585