The real-oper Hecke eigenvalue conjecture

Let $\mathcal C$ be a compact complex curve, let $G^{\vee}$ be the Langlands dual group, and let $o$ be a real $G^{\vee}$-oper. For a dominant coweight $\lambda$, let $V_\lambda$ be the corresponding representation, let $\sigma_\lambda$ be the canonical section arising from the oper structure, and let $(\cdot,\cdot)_\lambda$ be the pairing with the conjugate dual section. The real-oper Hecke eigenvalue conjecture. The scalar

$$\Phi_{\lambda,o}=(\sigma_\lambda,\overline{\sigma}_{-w_0(\lambda)})_\lambda$$

should be a smooth section of

$$|\omega_{\mathcal C}|^{-\langle\lambda,\rho^{\vee}\rangle}.$$

This proposes explicit geometric Hecke eigenvalues attached to real opers; the source gives no resolution status.