Monoidal abelian envelopes and a conjecture of Benson--Etingof

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite dimensional representations of $SL_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture.