The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension as a left module. Let $id(M)$ denote the injective dimension of a module $M$.

Question: Is there a general (noetherian if possible) ring $R$ with $id(R_R) \neq id(_{R}R)$?

I would expect that there is an easy counterexample or that this question has been considered before somewhere.