Classification of incompressible tensor categories of moderate growth in positive characteristic via Verlinde subcategories

Determine whether every incompressible tensor category of moderate growth over a field of positive characteristic is a tensor subcategory of a higher Verlinde category Ver_{p^n} (including Ver_{p^{\infty}}), as per the classification of subcategories given in Benson–Etingof [BEO].

Background

The authors discuss higher Verlinde categories Ver_{p^n} and their role in the structure theory of tensor categories of moderate growth. A result of Coulembier–Etingof–Ostrik shows that every pretannakian category of moderate growth admits a tensor functor to an incompressible category of moderate growth.

In positive characteristic, it is conjectured that incompressible categories of moderate growth are precisely subcategories of these Verlinde categories (including Ver_{p^{\infty}}), with subcategories classified in Benson–Etingof. This classification conjecture underpins broader structural expectations about tensor categories in positive characteristic.