Integral and compatible Satake parameters for discrete automorphic representations

Let $G$ be a connected reductive group over a global function field $K$, let $U\subset G(\mathbf A_K)$ be open compact, let $E$ be a noetherian $\mathbb Z[\frac1p]$-algebra embedded in an algebraically closed field $C$, and let ${{\mathbb C} A}_{disc}(G,U,E)$ be the discrete automorphic forms with coefficients in $E$. For a discrete automorphic representation belonging to a semisimple Langlands parameter $\sigma$, and for an unramified place $w\notin S(U)$, let $s_{w,\ell}(\sigma)$ be the Satake parameter corresponding to the Hecke character, let $\alpha_w(\sigma)=[\sigma(\mathsf{Frob}_w)]$, and let $\iota_\ell$ be the comparison embedding. Lafforgue's conjecture. (i) The discrete spectrum admits the excursion-operator decomposition described in the source. (ii) For fixed $\sigma$, there is an $s_w(\sigma)$ such that

$$s_{w,\ell}(\sigma)=\iota_\ell(s_w(\sigma)),\qquad \alpha_w(\sigma)=s_w(\sigma).$$

The source notes that this is known for cuspidal representations; the conjectural content concerns the discrete spectrum.