I'm using cohomological gradings. For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ induces a functor $\mathsf{Ho}(C\text-\mathsf{dg\text-mod})$ to $\mathsf{Ho}(A\text-\mathsf{dg\text-mod})$, where the modules are unbounded.

Is there a caracterisation of those $C$ such that the family of functors $$\{-\otimes^{\mathbb L}_CA\ |\ A\in k\text-\mathsf{cdga}_{\leq0},f:C\to A\}$$ is conservative, i.e. such that $M\in C\text-\mathsf{dg\text-mod}$ is uniquely determined (in the derived category) by the data of all those dg-modules $M\otimes^{\mathbb L}_C A$ for all connective commutative $k$-dg-algebra $A$ ?