Modular rigidity conjecture for cuspidal representations of ${\rm GL}_n$

Let $k$ be a global field, let $\ell$ be a prime different from the characteristic of $k$, and let $\iota:\mathbb{C}\to\overline{\mathbb{Q}}_\ell$ be a field isomorphism. Let $\Pi_1$ and $\Pi_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A})$ with central characters $\Omega_1$ and $\Omega_2$. Suppose that $\Omega_1\otimes_{\mathbb{C}}\overline{\mathbb{Q}}_\ell$ and $\Omega_2\otimes_{\mathbb{C}}\overline{\mathbb{Q}}_\ell$ are $\overline{\mathbb{Z}}_\ell$-valued and congruent modulo $\mathfrak{m}_\ell$, and that there is a finite set $S$ of places containing all Archimedean places and all finite places above $\ell$ such that, for every $v\notin S$, the local components are unramified and the characteristic polynomials of their Satake parameters lie in $\overline{\mathbb{Z}}_\ell[X]$ with the same reduction in $\overline{\mathbb{F}}_\ell[X]$. Let $w$ be a finite place not dividing $\ell$ such that the representations $\Pi_{1,w}\otimes_{\mathbb{C}}\overline{\mathbb{Q}}_\ell$ and $\Pi_{2,w}\otimes_{\mathbb{C}}\overline{\mathbb{Q}}_\ell$ are integral. Modular rigidity conjecture. The reductions modulo $\ell$ of these representations have a common generic irreducible component; this generic component is unique and occurs with multiplicity $1$. The conjecture predicts that global unramified congruence data, together with congruent central characters, rigidly determines the generic part of the relevant local reductions. Its status is not established by the supplied text.