Commutativity conjecture for the categorifying functors

Establish, for any distinct indices \(i,j,k\), an isomorphism of derived functors \(\mathcal{F}_{ij}\circ\mathcal{F}_{ik}\simeq\mathcal{F}_{ik}\circ\mathcal{F}_{ij}\).

Background

The conjecture proposes the categorical lift of the pairwise commutativity of the three genus-two Macdonald operators. The functors $\mathcal{F}_{ij}$ are defined from derived induction, restriction, and twisting by diagrammatic automorphisms associated with the degenerate affinization of $D(2|1,\alpha)$.