Oper formula for higher-degree affine Gaudin Hamiltonian eigenvalues

Let $d$ be a positive exponent with $d>1$, let $\gamma$ be a Pochhammer contour, and let $\Psi^{\mathrm{on}}_{\bm c}(\bm w)$ be an on-shell Bethe state in $H_{\bm\lambda}$. Associate to it the affine-oper class $[\nabla^{\bm c}_{\bm w}]$, whose quasi-canonical coefficient of exponent $d$ is $\mathcal{C}^d_{\bm c}(z,\bm w)$, and let $\mathscr{P}(z)$ be the twist-function primitive factor. Oper eigenvalue conjecture. The state $\Psi^{\mathrm{on}}_{\bm c}(\bm w)$ is an eigenvector of $\mathscr{H}^d_\gamma$, with eigenvalue

$$I^d_\gamma([\nabla^{\bm c}_{\bm w}])=\oint_\gamma \mathscr{P}(z)^{-d/h^\vee}\mathcal{C}^d_{\bm c}(z,\bm w)\,\mathrm{d}z.$$

This proposes a hypergeometric-integral description of the higher-degree Hamiltonian spectrum, extending the known quadratic Bethe-ansatz eigenvalue formula; it remains unproved in general.