Weak Ihara conjecture for cyclic automorphic modules

Let $F^+$, $G$, $U$, $v_0$, $\mathfrak{m}$, and $f$ be as in the weak Ihara lemma setup: $U\subset G(\mathbb{A}_{F^+}^{\infty})$ is sufficiently small, $v_0\in T-(S_{\ell}\cup S(B)\cup S_a)$ with $U_{v_0}\cong G(\mathcal{O}_{F_{v_0}^+})$, $\mathfrak{m}\subset\mathbb{T}^T(U)$ is non-Eisenstein, and $f\in S(U,k)[\mathfrak{m}]$. Consider the cyclic $k[G(F_{v_0}^+)]$-module

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

Weak Ihara conjecture. The $k[G(F_{v_0}^+)]$-module $\langle G(F_{v_0}^+)\cdot f\rangle$ is generic. This is explicitly presented as a weaker conjecture than the preceding weak Ihara lemma, since genericity of the whole module implies the desired genericity property for its irreducible submodules in the intended application. The source gives no evidence that this weaker conjecture has been resolved.