The existence of non-quasi-isomorphic DGAs equivalent as MU-algebras

Let $A$ and $B$ be differential graded algebras (DGAs), and let $\MU$ denote the complex cobordism spectrum. The MU-algebra equivalence conjecture. There exist DGAs $A$ and $B$ that are not quasi-isomorphic but are equivalent as $\MU$-algebras. This would give examples of DGAs distinguished by their algebraic homotopy type but identified by their $\MU$-algebra structure; the statement is presented as still open.