Emerton's conjecture on the cohomology of the modular tower

Let $L$ be the coefficient field, let $p$ be a prime, and let $\rho:{\cal G}_{\bf Q}\to {\bf GL}_2(L)$ be an odd representation. For each prime $\ell$, write $\rho_\ell$ for its restriction to ${\cal G}_{{\bf Q}_\ell}$. For $\ell\neq p$, let ${\rm LL}(\rho_\ell)$ be the representation of ${\mathbb G}({\bf Q}_\ell)$ attached to $\rho_\ell$ by the classical local Langlands correspondence, and let ${\rm LL}_p(\rho_p)$ be the representation of ${\mathbb G}({\bf Q}_p)$ attached to $\rho_p$ by the $p$-adic correspondence. Let $\widehat H^1_p$ be the $p$-adic completed cohomology representation introduced in the source.

Emerton's conjecture. If $\rho$ is unramified outside finitely many places, then

$${\rm Hom}_{{\cal G}_{\bf Q}}(\rho,\widehat H^1_p)\cong \Big(\bigotimes_{\ell\neq p}{\rm LL}(\rho_\ell)\Big)\otimes{\rm LL}_p(\rho_p).$$

In particular, ${\rm Hom}_{{\cal G}_{\bf Q}}(\rho,\widehat H^1_p)\neq 0$.

The conjecture describes the automorphic representation occurring in completed cohomology with Galois representation $\rho$. The source states that it has been largely proved by Emerton, but does not specify a complete resolution, so its database status remains open.