The universal Verlinde category conjecture for pre-Tannakian categories

Assume that $p>0$. A pre-Tannakian category is a $\boldsymbol{k}$-linear, rigid, symmetric monoidal abelian category with finite-dimensional Hom spaces and finite length objects. It is of subexponential growth if, for every object $X$, the length of $X^{\boxtimes n}$ is bounded by $a_X^n$ for some $a_X\in\mathbb{R}$. Let $\operatorname{Ver}_p$ denote the universal Verlinde category associated with the prime $p$. The universal Verlinde category conjecture. Every pre-Tannakian category $\mathcal{C}$ of subexponential growth admits a unique up to isomorphism $\boldsymbol{k}$-linear exact symmetric tensor functor

$$\mathcal{C}\longrightarrow \operatorname{Ver}_p.$$

This conjecture proposes a positive-characteristic extension of Deligne's characteristic-zero criterion for the existence of a super fiber functor. The paper introduces $\operatorname{Ver}_p$ as the proposed universal target, while the conjectural existence and uniqueness of the functor remain unresolved in the supplied text.