The local-global compatibility conjecture for the Hecke-algebra Galois representation

Let $E$ be the quadratic field under consideration, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, and write $\mathbf{T}:=h_\mathfrak{m}$. Let

$$\rho_\mathfrak{m}:{\mathrm{Gal}}(\overline{E}/E)\to\mathrm{GL}_n(\mathbf{T})$$

be the Galois representation in the preceding conjecture. Local-global compatibility conjecture. The representation $\rho_\mathfrak{m}$ satisfies local-global compatibilities at minimal, Fontaine–Laffaille, and Taylor–Wiles places. Some compatibilities are known for related representations over nilpotent quotients in the imaginary CM case, while the source states that no such compatibilities at bad places are known for the representation in the completely split case.