Generalized Emerton factorization conjecture for completed cohomology

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 a finite set $S$ of primes, write $\widehat H^1$ for the completed cohomology representation, let ${\rm LL}_p(\rho_\ell)$ denote the $p$-adic local Langlands representation at each $\ell\in S$, and let ${\mathbb G}({\bf A}_f^S)$ be the finite adèlic group away from $S$ and $p$ as defined in the source.

Generalized Emerton conjecture. For every finite set $S$, there is a factorization

$${\rm Hom}_{{\cal G}_{\bf Q}}(\rho,\widehat H^1)\cong \Big(\,\widehat{\bigotimes_{\ell\in S}}\,{\rm LL}_p(\rho_\ell)\Big)\,\widehat\otimes\Pi^S_p(\rho),$$

where $\Pi^S_p(\rho)$ is a continuous representation of ${\mathbb G}({\bf A}_f^S)$.

This is proposed as a generalization of the preceding Emerton conjecture, replacing the $p$-adic completed cohomology statement by a factorization valid for every finite set of places. The source presents it as a tentative generalization and gives no resolution, so it remains open.