Let $A$ be a connected finite dimensional basic algebra.
Question: Is there an indecomposable $A$-bimodule $W$ such that the finitistic dimension of $A$ is equal to the right projective dimension of $W$ and the finitstic dimension of $A^{op}$ is equal to the left projective dimension of $W$ and $W$ is additionally basic as a left and right module? Can we even have that $W$ is basic tilting as a left and right module?
The answer is positive for Gorenstein algebras, where on can choose $W=D(A)$ which is a basic tilting module.
The question just comes from curiousity without any good evidence that this could be true.
edit: In the original question I had just asked the question without assuming $W$ to be basic or tilting and then one can take $M \otimes_k N$ as Jeremy Rickard noted when $M$ has projective dimension equal to the finitistic dimension of $A$ and $N$ projective dimension equal to the finitistic dimension of $A^{op}$ .
