Affine Gaudin eigenvector conjecture for Schechtman–Varchenko vectors

Let $\boldsymbol{ abla}$ be a Miura oper for the Langlands dual affine Lie algebra on $\boldsymbol{ obreak\textrm{P}} obreak^1$, let $\boldsymbol{ obreak\boldsymbol{\theta}} obreak$ be the associated Schechtman–Varchenko vector in the specified weight space, and let $v_i(z)$ be the coefficient of the degree-$i$ generator in a quasi-canonical form of the underlying oper. Let the Bethe roots satisfy the Bethe equations, and let \boldsymbol{Q}^\boldsymbol{\beta}_i be the contour-integrated higher Hamiltonians.

Affine Gaudin eigenvector conjecture. For every exponent $i$ and every permitted contour $\boldsymbol{\beta}$,

\boldsymbol{Q}^\boldsymbol{\beta}_i\boldsymbol{\theta}=\left(\int_\boldsymbol{\beta}\boldsymbol{\rho}(z)^{-i/\nu}v_i(z)\,dz\right)\boldsymbol{\theta}.

Thus the Schechtman–Varchenko vector is a simultaneous eigenvector, with eigenvalues given by the hypergeometric contour integrals of the affine-oper coefficients. The statement depends on the Bethe equations and is presented as conjectural in the supplied text.