Equivalence of the semisimplification with the Verlinde quotient

Let $H_V$ be the reductive algebraic group and let $\overline{\mathcal{C}}_{\overline{V_\psi}}$ be the semisimplification of the tensor category generated by the corresponding object $\overline{V_\psi}$. For sufficiently large primes $p$, there is a surjective tensor functor

$$F:{\rm Ver}_p(H_V)\to \overline{\mathcal{C}}_{\overline{V_\psi}}.$$

Equivalence conjecture. For sufficiently large $p$, the functor $F$ is an equivalence.

The preceding theorem establishes that the target is a quotient of ${\rm Ver}_p(H_V)$ and, in particular, a fusion category; the conjecture asserts that this quotient has no further nontrivial kernel.