Let $(A,m,k)$ be a commutative local ring.
Assume that for all $n\ge 3$, the Andre-Quillen homology modules $H_n(A,k,k)$ vanish.
Does this imply that $A$ has finite injective dimension over itself?
If I knew that $A$ is noetherian then the answer would obviously be yes, because such a vanishing result implies that $A$ is a complete intersection ring, and hence, Gorenstein.
What about the general case?
