Concentration conjecture for localized completed homology

Let $U^p$ be a compact open subgroup away from $p$, let $\mathcal{O}$ be the coefficient ring, let $\mathbb{T}^S(U^p)$ be the Hecke algebra, and let $q_0$ be the distinguished homological degree. For a maximal ideal $\mathfrak{m}\subset\mathbb{T}^S(U^p)$, call $\mathfrak{m}$ non-Eisenstein when the associated representation $\bar{\rho}_{\mathfrak{m}}$ is absolutely irreducible. Concentration conjecture. If $\mathfrak{m}\subset\mathbb{T}^S(U^p)$ is a non-Eisenstein maximal ideal, then

$$\widetilde{H}_\ast(X_{U^p},\mathcal{O})_{\mathfrak{m}}=\widetilde{H}_{q_0}(X_{U^p},\mathcal{O})_{\mathfrak{m}}.$$

These conjectures are open in general, but are known to hold when $l_0=1$, namely when $F$ is an imaginary quadratic field.