For which commutative rings k is the following true:

A k-algebra $A$ that is flat over $k$ and derived equivalent to a $k$-algebra $B$ implies that also $B$ is flat over $k$.

For which commutative rings k is the following true:

A k-algebra $A$ that is flat over $k$ and derived equivalent to a $k$-algebra $B$ implies that also $B$ is flat over $k$.

The motivation is this: Many results on derived equivalences start with two k-algebras A and B that are assumed to be flat and derived equivalent. Maybe it is often enough to just assume one is flat?