Remarks on $q$-difference opers arising from quantum toroidal algebras

We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal ${\mathfrak{g}\mathfrak{l}}_2$ 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 ${\mathfrak{g}\mathfrak{l}}_n$.