Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero?

In case this is true, this might generalise the result that selfinjective algebras are closed under derived equivalence. (Since selfinjective algebras have finitistic dimension zero and they are exactly the Gorenstein algebras with finitistic dimension zero (and being Gorenstein is preserved under derived equivalence)).

edit: It might be a better question to ask for the two-sided condition. That is: Let $A$ be an Artin algebra of left and right finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have left and right finitistic dimension zero?

I think in this form it might be true. Non-selfinjective examples with left and right finitistic dimension zero are quiver algebras where at each vertex there is at least one loop.