Stronger Ihara's lemma for unitary groups

Let $G$, $F^+$, $F$, $T$, $S_{\ell}$, $S(B)$, $U^v$, $S_{a,\{\rho_x\},\varnothing}$, and $\bar{k}$ have the meanings used in the paper. Let a maximal ideal be non-Eisenstein when its associated residual representation $\bar{r}_{\mathfrak m}$ is absolutely irreducible. Stronger Ihara's lemma. Let $U\subset G(\mathbb A_{F^+}^{\infty})$ be sufficiently small, let $v\in T-(S_{\ell}\cup S(B))$ be split in $F$ with no restriction on $U_v$, and let $\mathfrak m\subset\mathbb T_{a,\{\rho_x\},\varnothing}^T(U)$ be non-Eisenstein. Then every irreducible $\bar{k}[G(F_v^+)]$-submodule

$$\pi\subset S_{a,\{\rho_x\},\varnothing}(U^v,\bar{k})_{\mathfrak m}$$

is generic. This strengthens the preceding Ihara-lemma assertion by removing the cyclic-submodule and spherical-eigenform hypotheses; the source says it is not needed for the argument and records it as a stronger conjecture.