Liftability of stable Picard group orbits of simple-images

Determine whether every stable Picard group orbit of simple-images of Morita type for a self-injective algebra contains a representative that is liftable to a derived equivalence.

Background

A simple-image of Morita type is obtained by transporting the non-projective simple modules of one algebra across a stable equivalence of Morita type. Such an image is liftable if the stable equivalence can be realized as the stable restriction of a derived equivalence while preserving the specified simple-image. The paper formulates the orbit-level lifting question for arbitrary self-injective algebras, notes that it has a negative answer in general, and proves a positive answer for Brauer graph algebras.

References

Let $A$ be a self-injective algebra. Does every $\operatorname{StPic}(A)$-orbit of simple-images of Morita type contain a liftable representative?

— Classification of Brauer graph algebras under stable equivalence of Morita type  (2608.22715 - Li et al., 24 Aug 2026) in Problem 1, Section 1, Introduction