The Galois representation conjecture for Hecke algebras

Let $E$ be the quadratic field under consideration, let $h$ be the Hecke algebra acting on the relevant cohomology, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, and write $\mathbf{T}:=h_\mathfrak{m}$. Let $S$ be the finite set of places outside which the residual Galois representation is unramified, and let $q_w$ and $T_{w,i}$ denote the quantities occurring in the Hecke polynomial at a place $w\notin S$. Galois representation conjecture. There exists a Galois representation

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

unramified outside of $S$ and such that, for each $w\notin S$, the characteristic polynomial of $\rho_\mathfrak{m}({\mathrm{Frob}}_w)$ is

$$\sum_{i=0}^n(-1)^i q_w^{i(i-1)/2}T_{w,i}X^{n-i}\in\mathbf{T}[X].$$

Scholze proves a version after quotienting by a nilpotent ideal; further results establish variants in some settings, including when $p$ is completely split in $E$, but the full assertion remains open in the generality stated.