Calabi-Yau completions and orbifold equivalences

Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth dg algebras also form a 2-category to which the construction of `equivariant completion' can be applied. In this theory two objects are called `orbifold equivalent' if there is a 1-morphism $X$ with invertible quantum dimensions between them. Any such relation entails a whole family of equivalences between categories. We show that an orbifold equivalence between two homologically smooth and proper dg algebras lifts to an orbifold equivalence between their Calabi-Yau completions under certain conditions on $X$.