Main K-theoretic convolution equivalence conjecture

Let $G$ be the group under consideration, $G^\vee$ its Langlands dual group, $KConv^{\widetilde{G^\vee}}(Gr)$ and $KConv^{\widetilde{G^\vee}\times\mathbb C^\times}(Gr)$ the equivariant K-theory convolution categories, and let $E$, $E^\vee$, $q$, $\mathcal O^{\min}(G/G)\operatorname{-mod}$, and $\mathcal O_q^{\min}(G/G)\operatorname{-mod}$ have the meanings fixed in the paper. The categories are enriched over the indicated module categories, and objects of length $m$ carry actions of $\mathbb Z^m$. Main K-theoretic convolution equivalence conjecture. There are equivalences

$$\mathcal O^{\min}(G/G)\operatorname{-mod}\otimes_E E^\vee\cong KConv^{\widetilde{G^\vee}}(Gr)$$ $$\mathcal O_q^{\min}(G/G)\operatorname{-mod}\otimes_E E^\vee\cong KConv^{\widetilde{G^\vee}\times\mathbb C^\times}(Gr)\otimes_{\mathbb C[q^{\pm}]}\mathbb C(q),$$

of $E^\vee\operatorname{-mod}$- and $E^\vee(q)\operatorname{-mod}$-enriched categories, respectively. The equivalences should also be compatible with the actions of $\mathbb Z^m$ on objects of length $m$. This is the paper's main conjectural identification between K-theoretic convolution and quantum-equivariant module categories; it remains unproved.