Conjectural description of the isotypic part of cohomology of Shtukas

Let $C$ be the underlying smooth projective curve, let $I$ be a finite index set with $|I|=r$, and let $fs_n$ be a geometrically irreducible Weil local system of rank $n$ on $C$. Write $fs_n^{\underline{\e}}=\boxtimes_{i\in I}fs_n^{\e_i}$ for $\underline{\e}\in\{\pm1\}^r_0$, and let $\Sht_{\GL_n,\underline{\e}}$ denote the corresponding Shtuka stack. The notation $\Gamma_c(\Sht_{\GL_n,\underline{\e}},\underline{\Qlbar})_{\Loc_{\GL_n,fs_n}^{\arith}}$ denotes the isotypic summand supported on the connected component $\Loc_{\GL_n,fs_n}^{\arith}$. Conjectural description. For each $\underline{\e}\in\{\pm1\}^r_0$, there is a canonical isomorphism

$$\Gamma_c(\Sht_{\GL_n,\underline{\e}},\underline{\Qlbar})_{\Loc_{\GL_n,fs_n}^{\arith}}[(n-1)r]\cong \Gamma(C^I,fs_n^{\underline{\e}})\otimes \cO(\Loc_{\GL_n,fs_n}^{\arith}).$$

In particular,

$$(\prod_{d\in\ZZ}H^{*+(n-1)r}_c(\Sht_{\GL_n,\underline{\e}}^d,\underline{\Qlbar}))_{fs_n}\cong H^*(C^I,fs_n^{\underline{\e}}).$$

This is presented as a conjectural description of the geometric-isotypic contribution to Shtuka cohomology and is related in the source to geometric Langlands with restricted variation; no resolution status is supplied.