Let $A$ be a finite dimensional algebra over a field $K$ given by quiver with relations and assume for simplicity that it has finite global dimension. It is well known that the projective dimension of the regular module as a bimodule equals the global dimension of the algebra. Let $D(A):=Hom_K(A,K)$ the dual of the regular module, which is also a bimodule. It seems that it is not known what the projective dimension of $D(A)$ is in general.

Do we have that the projective dimension of $D(A)$ as an bimodule is larger than or equal to $\max \{ projdim(S)+injdim(S) | S$ is a simple right A-module $ \}$ ?

It is true for all Nakayama algebras with a linear quiver and at most 6 simple modules. Surprisingly, we have equality very often, namely of all 64 Nakayama algebras with a linear quiver with at most 6 simple modules, only the algebra with Kupisch series [ 3, 4, 3, 3, 2, 1 ] had that the projective dimension of $D(A)$ as a bimodule was 4 while $\max \{ projdim(S)+injdim(S) | S$ is a simple right A-module $ \}$=3 and for the other 63 algebras we had equality of the two numbers.