Eigenvalue formula conjecture for the first Hecke operator of $SL_n$

Let $\chi$ be an $SL_n$-oper with real monodromy. Let $(\mathcal V_{\omega_1},\nabla_\chi)$ be its associated flat vector bundle, with determinant identified with the trivial flat line bundle, and let $s_{\omega_1}$ and $s_{\omega_{n-1}}$ be the canonical oper sections in the associated bundles. Let $\{\Phi_\chi^\epsilon\}$ denote the $\mu_n$-torsor of eigenvalues of the first Hecke operator. First Hecke eigenvalue formula conjecture.

$$\{\Phi_\chi^\epsilon\}=\{\epsilon h_\chi(s_{\omega_1},\overline{s_{\omega_{n-1}}})\}$$

as $\mu_n$-torsors of global sections of $\Omega_X^{-(n-1)/2}$. This gives an explicit geometric expression for Hecke eigenvalues in terms of the flat pairing induced by real monodromy; the formula is conjectural in the stated generality.