The conjecture on cohomology for minimal elliptic pairs

Let $(T,U)$ be a minimal elliptic pair, and let $\chi\colon \mathcal{T}^{+}(\mathcal{O}_k)\to\overline{\mathbb{Q}}_\ell^{\times}$ be a smooth character. Let $s_\chi\geq 0$ denote the degree in which the homology of $f_\natural\overline{\mathbb{Q}}_\ell[\chi]$ is non-vanishing, when such a degree is specified by Theorem 1. The associated cohomology representations and Frobenius action are those described in the three assertions of Theorem 1. Conjecture on minimal elliptic pairs. Theorem 1 holds for all minimal elliptic pairs $(T,U)$. The conjecture extends the theorem's conclusions from Coxeter pairs to minimal elliptic pairs; the supplied text states that part (3) is already known for minimal elliptic pairs, while the full assertion remains unresolved.