Ihara's lemma for generic irreducible submodules

Let $F^+$ be the totally real field in the global setup, let $G$ be the associated group, and let $U\subset G(\mathbb{A}_{F^+}^{\infty})$ be sufficiently small. Let $v_0\in T-(S_{\ell}\cup S(B)\cup S_a)$ satisfy $U_{v_0}\cong G(\mathcal{O}_{F_{v_0}^+})$, let $\mathfrak{m}\subset\mathbb{T}^T(U)$ be a non-Eisenstein maximal ideal, and let $f\in S(U,k)[\mathfrak{m}]$. If $\pi$ is an irreducible $k[G(F_{v_0}^+)]$-submodule of

$$\langle G(F_{v_0}^+)\cdot f\rangle\subset S(U^{v_0},k),$$

Ihara's lemma. Then $\pi$ is generic. This is the precise weak Ihara lemma attributed in the source to Clozel, Harris, and Taylor. The introduction explains that the corresponding Ihara conjecture is known for $n=2$ but remains open for $n>2$; no resolution is supplied here.