Benson–Etingof–Ostrik conjecture on fiber functors to ${\rm Ver}_{p^\infty}$

Let $\mathbf{k}$ be an algebraically closed field of characteristic $p>0$. A symmetric tensor category of moderate growth is a symmetric tensor category over $\mathbf{k}$ satisfying the moderate-growth condition. Let

$${\rm Ver}_{p^\infty}=\bigcup_{n\geq 1}{\rm Ver}_{p^n},$$

where ${\rm Ver}_{p}\subset {\rm Ver}_{p^2}\subset\cdots$ is the nested sequence of incompressible symmetric tensor categories constructed in the paper. Benson–Etingof–Ostrik conjecture. Any symmetric tensor category of moderate growth over $\mathbf{k}$ admits a fiber functor to ${\rm Ver}_{p^\infty}$. This predicts that ${\rm Ver}_{p^\infty}$ is universal for fiber functors from symmetric tensor categories of moderate growth in positive characteristic; the conjecture is presented as a proposed, currently unresolved classification statement.