Ihara's lemma in the bottom homological degree

Let $K$ and $v$ be as in the paper's level-lowering setup, and let $\mathfrak{m}\subseteq\mathbb{T}^S(K)$ be a non-Eisenstein maximal ideal. Let

$$\pi_{K,v}\colon (C_{K\cap K_0(v)})_{\mathfrak{m}}\to(C_K)_{\mathfrak{m}}^{\oplus2}$$

be the level-lowering map. Ihara's lemma conjecture. The induced map

$$\operatorname{H}_{r_1+r_2}(Y_{K\cap K_0(v)},\mathcal{O})_{\mathfrak{m}}\to\operatorname{H}_{r_1+r_2}(Y_K,\mathcal{O})_{\mathfrak{m}}^{\oplus2}$$

is surjective. This is the higher-dimensional analogue of Ihara's lemma and is used in level raising and modularity-lifting arguments; the supplied text gives no resolution status.