Koszul duality conjecture for mirror dual symplectic singularities

Let the Higgs and Coulomb branches of a 3d $\mathcal{N}=4$ supersymmetric gauge theory be mirror dual symplectic singularities, and let $\mathcal{O}$ denote the category associated to a symplectic singularity with a $\mathbb{C}^*$-action. Koszul duality conjecture. The categories $\mathcal{O}$ of mirror dual symplectic singularities are Koszul dual. This extends the Koszul duality phenomena known for categories such as singular and parabolic blocks of category $\mathcal{O}$ to Higgs–Coulomb mirror symmetry; the general claim is presented as a conjecture based on computations in numerous examples, and its status is not resolved here.