Ihara lemma, indefinite case

Assume the indefinite Shimura-variety setup of the source. Let $\mathcal{S}_0^\circ$ and $\mathcal{S}_0^\bullet$ be the relevant Shimura varieties, let $\Omega_\lambda$ be the local system, and let $\mathfrak{n}_0$ denote the localization ideal. Ihara lemma, indefinite case. If $\ell\nmid p\prod_{i=1}^{n_0}(1-(-p)^i)$, if $\mathrm{H}^i(\mathcal{S}_0^\circ\otimes_F\overline{\mathbb{Q}},\mathbb{Z}_\lambda)_{\mathfrak{n}_0}=0$ for $i\neq n-1$ and $\mathrm{H}^{n-1}(\mathcal{S}_0^\circ\otimes_F\overline{\mathbb{Q}},\mathbb{Z}_\lambda)_{\mathfrak{n}_0}$ is finite free over $\mathbb{Z}_\lambda$, if $\mathrm{V}^\Pi_{0,\lambda}$ is residually absolutely irreducible, and if the Satake parameter of $\Pi_{0,\mathfrak{p}}$ modulo $\lambda$ contains $p$ at most once, then

$$\mathrm{H}^{2r-1}(\mathcal{S}_0^\circ\otimes_F\overline{\mathbb{Q}},\Omega_\lambda)_{\mathfrak{n}_0}\longrightarrow \mathrm{H}^{2r-1}(\mathcal{S}_0^\bullet\otimes_F\overline{\mathbb{Q}},\mathbb{Z}_\lambda)_{\mathfrak{n}_0}$$

is surjective. This is stated as a generalization of the Ihara lemma for modular curves, or unitary Shimura curves, in the source.