Radical-cube-zero case

Determine whether every self-orthogonal -tilting module over an Artin algebra with radical cube zero is a classical 1-tilting module.

Background

The paper establishes that the self-orthogonal -tilting conjecture holds for radical square zero Artin algebras. It then examines algebras satisfying $\operatorname{rad}^3 A=0$ and proves that the endomorphism algebra associated with a self-orthogonal -tilting module has finite little finitistic dimension.

For the radical-cube-zero case, the paper reduces the remaining issue to forcing the annihilator into $Fac\,T$. The radical-square-zero argument does not directly extend because composition factors can occur in the middle Loewy layer rather than in the top, and the authors explicitly state that the general conclusion is unknown.