Cohomology rings of higher Verlinde categories

We prove the Benson–Etingof conjecture describing the unit cohomology rings of the higher Verlinde categories \(\mathrm{Ver}_{p^{n+1}}\) over algebraically closed fields of positive characteristic. The resulting isomorphisms preserve the standard parameters and are compatible with the inclusions of the Verlinde tower. The proof combines exact transport of rank-one costandard sequences, a description of a regular idempotent corner as a tensor product, and explicit comparisons of coefficient classes through truncated Yoneda extensions. These comparisons determine the full image in a localized cohomology ring, including nilpotent classes, and permit exact descent of multiplication. Global freeness over the standard polynomial parameter subalgebra is established before localization.