Conjecture 4.2: Correspondence between eigenvectors and q-hypergeometric opers for evaluation relaxed Verma modules

Establish a one-to-one correspondence between joint eigenvectors of the Bethe algebra A(p0, p1) acting on the degree (0, −l0) subspace of the evaluation relaxed Verma module Gµ,ν of the quantum toroidal gl2 algebra and the set of second-order q-difference opers L of the explicit form given in equation (4.10), subject to the apparent singularity conditions (4.11) and the linear relations (4.12) with parameters λ1 and x1 defined by equation (4.9).

Background

Building on the treatment of evaluation Verma modules, Section 4.2 considers evaluation relaxed Verma modules Gµ,ν obtained by induction and evaluation from Uq(gl2). The authors predict that the spectral problem for the Bethe algebra can be reformulated via specific second-order q-opers with prescribed apparent singularities and algebraic constraints.

Conjecture 4.2 provides a detailed oper-theoretic classification of spectra in relaxed settings, including an explicit parametric form for the q-oper and constraints ensuring apparent singularities, thereby extending the oper correspondence in the quantum toroidal framework.