Nakayama's conjecture for finite-rank algebras

Let $A$ be a not necessarily commutative algebra of finite rank over a field $k$, and let $A$ be viewed as a left $A$-module. Nakayama's conjecture. If $A$ has an injective resolution as a left $A$-module in which every term is also projective, then $A$ is selfinjective. This remains one of the main open problems in the representation theory of Artin algebras and is equivalent to formulations due to Tachikawa.