Ihara's lemma for unitary groups

Let $G$ be the unitary group and retain the paper's notation for the coefficient system $S_{a,\{\rho_x\},\varnothing}$, the set of places $T$, and the level subgroup $U$. A place $v$ is banal in the sense used in the paper, and $U^v$ denotes the corresponding level modification. Let $\bar{k}$ be the residue field, and call a maximal ideal non-Eisenstein when its associated residual representation $\bar{r}_{\mathfrak m}$ is absolutely irreducible. Ihara's lemma. Let $U\subset G(\mathbb A_{F^+}^{\infty})$ be sufficiently small, let $v\in T-(S_{\ell}\cup S(B))$ be a place where $U_v=G(\mathcal O_{F_v^+})$, let $\mathfrak m\subset\mathbb T_{a,\{\rho_x\},\varnothing}^T(U)$ be a non-Eisenstein maximal ideal, and let $f\in S_{a,\{\rho_x\},\varnothing}(U,\bar{k})[\mathfrak m]$ be an eigenform. Then every irreducible $\bar{k}[G(F_v^+)]$-submodule $\pi$ satisfying

$$\pi\subset\langle G(F_v^+)f\rangle\subset S_{a,\{\rho_x\},\varnothing}(U^v,\bar{k})$$

is generic. This is the Ihara-lemma input for the non-minimal modularity-lifting argument; the paper notes that it follows from the trivial-data case, but treats the assertion as conjectural here.