  The finitistic dimension conjecture is closely connected to the symmetry of
the finitistic dimension. Recent work indicates that such connection extends to
one of its upper bounds, the delooping level. In this paper, we show that the
same holds for the derived delooping level, which is an improvement of the
delooping level. This reduces the finitistic dimension conjecture to
considering algebras whose opposite algebra has (derived) delooping level zero.
We thereby demonstrate ways to utilize the new concept of derived delooping
level to obtain new results and present additional work involving tensor
product of algebras.
