Ihara's Lemma for $\mathrm{GL}_n$: the limit case

Clozel, Harris and Taylor proposed conjectural generalizations of the classical Ihara's lemma for $\mathrm{GL}_2$, to higher dimensional similitude groups. We prove these conjectures in the so called limit case, which after base change is the essential one, under any hypothesis allowing level raising as for example in the work of Gee.