The equivariant bundle conjecture for Hecke operators

Let $C/\mathbb F_q$ be the fixed curve and let $X$ and $F:X\to X$ be as in the first dynamical proposal. For $x\in C(\mathbb F_q)$, let $T_x^{(n)}$ be the corresponding Hecke operator over $\mathbb F_{q^n}$, and let $X_n$ denote the corresponding set of fixed points. An $F$-equivariant vector bundle is a vector bundle $\mathcal E$ on $X$ together with an isomorphism $g:F^*\mathcal E\to\mathcal E$.

Equivariant bundle conjecture. There exists an $F$-equivariant vector bundle $(\mathcal E,g)$ of rank $N$, defined over $\mathbb Q$, such that for $z\in X_n$ the eigenvalue of $T_x^{(n)}$ at the spectral point corresponding to $z$ equals

$$\operatorname{Trace}\left(\mathcal E_z=\mathcal E_{F^n(z)}\to\cdots\to\mathcal E_{F(z)}\to\mathcal E_z\right),$$

where the arrows are the fiber isomorphisms induced by $g$.

This conjecture packages the spectra of the Hecke operators over all extensions into the trace of monodromy of one equivariant bundle. It depends on the preceding dynamical realization and is not proved in the source.