The stable-category form of the Tachikawa conjecture

Let $A$ be a finite-dimensional self-injective algebra, and let $\underline{\mathsf{mod}}A$ denote its stable module category. The Tachikawa conjecture. The stable category $\underline{\mathsf{mod}}A$ has no non-trivial presilting objects. This is stated as equivalent to the module-theoretic Tachikawa conjecture, which asserts that a module with vanishing positive self-extensions is projective; the source does not state whether it is resolved.