Knutson-type conjecture for complex finite-dimensional group algebras

Let $G$ be a finite group, and let $\mathbb{C}[G]$ be its complex group algebra. A bialgebra is of Knutson type if every simple module over it has trivial Knutson index, meaning that each simple module $M$ has a virtual module $N$ with $M\otimes N\cong A$. Knutson-type conjecture. All complex finite-dimensional group algebras are of Knutson type. This is a reformulation of Knutson's conjecture, but the paper revisits Savitskii's counterexamples and shows that the assertion does not hold.