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We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal gl₂ algebra on relaxed Verma modules, and q-hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear q-difference operators and discuss some of their properties. We also touch on a generalization to glₙ.
Feigin et al. (Tue,) studied this question.