Ihara's lemma conjecture for Shimura-curve cohomology

Let $q$ be a prime such that $q\nmid N\Delta'\ell^2$. Let ${\bf T}_0^{\widehat\psi}(N)$ act on $H^1({\bf X}_1(N),\mathcal O)^{\widehat\psi}$, and fix a maximal non-Eisenstein ideal $\mathfrak m$. Define the Shimura curve

$${\bf X}_1(N)=B^\times\setminus B_{\bf A}^\times/K_\infty^+V_1(N),$$

where

$$V_1(N)=\prod_{p\nmid N\ell}R_p^\times\prod_{p\mid N}K_p^1(N)\times(1+u_\ell R_\ell).$$

Let $\mathfrak m^q$ be the inverse image of $\mathfrak m$ under ${\bf T}_0^{\widehat\psi}(Nq)\to{\bf T}_0^{\widehat\psi}(N)$, and let $k=\mathcal O/\lambda$. Ihara's lemma conjecture. The map

$$\alpha_{\mathfrak m}:H^1({\bf X}_1(N),\mathcal O)_{\mathfrak m}^{\widehat\psi}\times H^1({\bf X}_1(N),\mathcal O)_{\mathfrak m}^{\widehat\psi}\longrightarrow H^1({\bf X}_1(Nq),\mathcal O)_{\mathfrak m^q}^{\widehat\psi}$$

is such that $\alpha_{\mathfrak m}\otimes_{\mathcal O}k$ is injective. This is the quaternionic analogue of Ihara's lemma needed to extend the preceding minimal-level results to higher level; the paper explicitly presents the assertion as a conjectural input from which the preceding conjecture would follow, so its resolution is not supplied here.