Categorical realization of all T-indecomposable inductive systems

Establish a bijection between T-indecomposable inductive systems for the finitary symmetric group algebra and isomorphism classes of semisimple objects in the symmetric monoidal category $Ver_{p^infty}$, as predicted by the cited conjecture.

Background

The paper relates ideals and inductive systems for the finitary symmetric group algebra to tensor ideals and objects in symmetric monoidal categories. The higher Verlinde category $Ver_{p^\infty}$ is proposed as the categorical setting that should encode all T-indecomposable inductive systems. The cited conjecture predicts a bijective correspondence, but the paper establishes only selected cases, including all systems of length at most three and certain minimal systems.

This problem is included because the paper explicitly presents its results as partial progress toward the broader categorical classification. In particular, proving the conjecture would provide a systematic classification of T-indecomposable inductive systems beyond the cases treated in the paper.