The classification conjecture for incompressible moderate-growth categories

Let $\mathbf{k}$ be an algebraically closed field of characteristic $p>0$, and let $\mathsf{Ver}_{p^\infty}$ denote the union of the higher Verlinde categories. A pretannakian category is incompressible if every tensor functor out of it is an embedding of a tensor subcategory; it has moderate growth if the lengths of tensor powers of each object are bounded exponentially.

Classification conjecture. Every incompressible category of moderate growth over $\mathbf{k}$ is a tensor subcategory of $\mathsf{Ver}_{p^\infty}$.

Together with the paper's theorem producing an incompressible moderate-growth target, this is equivalent to the higher Verlinde target conjecture. Tensor subcategories of the higher Verlinde union are classified in the cited work, but it is open whether all incompressible moderate-growth categories occur there.