Degeneration of quantum difference operators to quantum multiplication for Hilbert schemes of A_{r-1}-singularities

Prove that, for the equivariant Hilbert scheme Hilb_n(\widehat{C}^2/\mathbb{Z}_r) of A_{r-1}-singularities, the quantum difference operator M_{L_i}(z), taken in the eigenbasis H of M_{L_i}(\infty), degenerates to the quantum multiplication operator by the divisor line bundle L_i in the equivariant cohomology H_T^*(Hilb_n(\widehat{C}^2/\mathbb{Z}_r)), equivalently establishing the identification up to conjugacy required for the QDE-to-Dubrovin degeneration in this setting.

Background

The authors discuss known formulas for Dubrovin (quantum) connections on Hilbert schemes of A_{r-1}-resolutions and relate them to quantum difference equations. They formulate a specific conjectural identification for Hilb_n(\widehat{C}^2/\mathbb{Z}_r): that the quantum difference operators associated to divisors should degenerate to quantum multiplication by those divisors.

They note complementary geometric and algebraic viewpoints: via elliptic stable envelopes, the connection matrix should match the elliptic geometric R-matrix, while algebraically one needs to identify M_{L_i}(z) with the line bundle L_i up to conjugacy. They explicitly state that they do not yet know how to prove this identification and leave it for future work.