C-group Galois representation conjecture for Hecke eigenvalues

Let $\mathbb{T}$ be the Hecke algebra, let $x:\mathbb{T}[1/p]\to\overline{\mathbb{Q}}_p$ be a continuous homomorphism, and for each prime $\ell\notin S$ let $x_\ell:\mathcal{H}_\ell\to\overline{\mathbb{Q}}_p$ be the induced Hecke homomorphism. Let $\mathrm{CC}(x_\ell)$ be the semisimple conjugacy class obtained from $x_\ell$ through the $C$-group Satake isomorphism, let $d:{}^CG_f\to\mathbb{G}_m$ be the natural character, and let $\xi:\mathbb{G}_m\to\widehat{G}^T$ be

$$\xi(t)=((2\delta)(t^{-1}),t^2),$$

where $2\delta$ is the sum of the positive roots.

C-group Galois representation conjecture. There exists an admissible representation

$$\rho:\operatorname{Gal}_{\mathbb{Q}}\longrightarrow{}^CG_f(\overline{\mathbb{Q}}_p)$$

such that $d\circ\rho$ is the cyclotomic character, $\rho$ is unramified outside $S$, and for every $\ell\notin S$ the semisimplification of $\rho(\operatorname{Frob}_\ell)$ belongs to

$$\mathrm{CC}(x_\ell)\xi(\chi_{\mathrm{cyc}}(\operatorname{Frob}_\ell))\subset\widehat{G}(\overline{\mathbb{Q}}_p)\rtimes(\{\ell^{-1}\}\times\{\operatorname{Frob}_\ell\}).$$

This is an optimistic generalization of the conjectural association between Hecke eigenvalues and Galois representations, using the $C$-group Satake isomorphism. In the general setting of the paper, existence remains conjectural; under additional automorphic hypotheses it is related to the cited conjecture of Beijer, Gee and Geraghty.