On modular rigidity for ${\rm GL}_n$

Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $\Pi_1$ and $\Pi_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A}_k)$ for some integer $n\geq1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components $\Pi_{1,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $\Pi_{2,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components $\Pi_{1,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $\Pi_{2,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on $\Pi_1$ and $\Pi_2$, the answer is yes. We also give a simple proof when $k$ is a function field.