The homotopy Barr–Beck conjecture for DG-categories

Let $A_1$ and $A_2$ be DG-categories, and let $F$ and $G$ be a pair of homotopy adjoint functors between them. Let $T$ denote the homotopy monad induced by the composite of the adjunction data, and let $T$-$\operatorname{hoMod}(A_1)$ be the DG-category of strong homotopy modules over $T$ in $A_1$. Homotopy Barr–Beck conjecture. Under certain assumptions, the DG-categories $T$-$\operatorname{hoMod}(A_1)$ and $A_2$ are quasi-equivalent. This is intended as a DG-categorical analogue of the Barr–Beck theorem, which identifies a category with the category of modules over the monad induced by an adjunction. The source states the result only under unspecified assumptions, and no resolution status is given.