Alperin–Auslander conjecture on stable equivalences

Let $A$ and $B$ be finite-dimensional algebras over an algebraically closed field, and suppose they are stably equivalent, meaning their stable module categories are equivalent. Alperin–Auslander conjecture. If $A$ and $B$ are stably equivalent, then they have the same number of isomorphism classes of simple non-projective modules. Stable equivalence is weaker than Morita, Rickard, and derived equivalence, so this conjecture seeks a numerical invariant preserved at the stable level. The source presents it as a classical outstanding conjecture.