The conjecture on Galois representations attached to automorphic representations of GSp4

Let $G$ be the group under consideration, let $F$ be the underlying number field, and let $\pi=\pi_\infty\otimes\pi_f$ be an automorphic representation of $G(\mathbb A)$. For a split prime $l=xx'$ of $F$, let $K_l$ be the specified hyperspecial subgroup, let $T_{l,i}$ for $i\in\{0,1,2\}$ be the corresponding Hecke operators, and let $c_{l,i}$ be the eigenvalue of $T_{l,i}$ on $\pi_l^{K_l}$. If $\pi_\infty\simeq W(\lambda)$, write $\lambda$ in terms of integers $h_1>h_2>h_3>h_4$ satisfying $h_1+h_4=h_2+h_3$. Galois-representation conjecture. Up to isomorphism, there exists a unique continuous semisimple Galois representation with $\mathrm{GSp}_4$-structure

$$\rho_\pi:\mathrm{Gal}_F\rightarrow\mathrm{GSp}_4(\overline{\mathbb Q}_p)$$

satisfying the following conditions. If $l=xx'$ splits in $F$ and $\iota_x^*\pi_l$ is unramified, then $\rho_\pi$ is unramified above $l$, and

$$\imath^{-1}\mathrm{det}(T\mathrm{id}-\rho_\pi(\mathrm{Frob}_x))=T^4-c_{l,1}T^3+((l^3+l)c_{l,0}+lc_{l,2})T^2-l^3c_{l,0}c_{l,1}T+l^6c_{l,0}^2$$

and

$$\imath^{-1}\mathrm{sim}(\rho_\pi(\mathrm{Frob}_x))=l^3c_{l,0}.$$

If $l=xx'$ splits in $F$, then

$$\imath^{-1}\mathrm{WD}(\rho_\pi\vert_{\mathrm{Gal}_{F_x}})$$

coincides with $\iota_x^*(\pi_l\otimes\vert\mathrm{sim}\vert^{\frac32})$ under the classical local Langlands correspondence. If $p=\wp\wp'$ splits in $F$, then $\rho_\pi\vert_{\mathrm{Gal}_{F_\wp}}$ is de Rham and

$$\imath^{-1}\mathrm{WD}(\rho_\pi\vert_{\mathrm{Gal}_{F_\wp}})$$

coincides with $\iota_\wp^*(\pi_p\otimes\vert\mathrm{sim}\vert^{\frac32})$ under the classical local Langlands correspondence. Moreover, if $\pi_\infty\simeq W(\lambda)$, then $\rho_\pi\vert_{\mathrm{Gal}_{F_\wp}}$ has Hodge–Tate weights $h_1>h_2>h_3>h_4$. The conjecture predicts a compatible Galois representation realizing the local Langlands parameters and Hodge–Tate weights of automorphic representations of $G$; the supplied text does not state a resolution.