Mixed-category formulation of the Iwahori Gaiotto conjecture

Establish the equivalence between the category of (H(F),psi)-equivariant twisted D-modules on the affine flag variety Fl_G and the category of modules \cO_q^{\mxd}(\fg_s) for the mixed quantum supergroup, with positive Lusztig half and negative De ConciniKac half.

Background

The Iwahori Gaiotto conjecture is the quantum, representation-theoretic version of the tamely ramified relative Langlands conjecture. The mixed quantum supergroup is defined using a relative Drinfeld center, so its representation category can be formulated without first constructing the full Lusztig quantum supergroup.

Although the paper later proves this equivalence under Conditions (avoids small torsion) and (*) in the stated homogeneous cases, the passage itself records the conjecture that motivates the result; it is included here as an explicitly stated conjectural problem.