Rigidity and absolute irreducibility of reductions of automorphic Galois representations

Let $F$ be the number field in the setup, let $\Pi$ be a $\mathfrak d$-REASDC representation of $\operatorname{GL}_{2m}(\mathbb A_F)$, let $E$ be a strong coefficient field of $\Pi$, and for each finite place $\lambda$ of $E$ let

$$\rho_{\Pi,\lambda}:\operatorname{Gal}_F\to\operatorname{GL}_{2m}(E_\lambda)$$

be the associated continuous homomorphism. Let $\mathcal P^\Pi$ be the finite set of finite places outside which $\Pi$ is unramified. Fix a finite set $\mathcal P$ of finite places of $F$ containing $\mathcal P^\Pi$. Rigidity and irreducibility conjecture. Suppose that the base change of $\Pi$ to $F(\sqrt{\mathfrak d})$ is also cuspidal. Then for all but finitely many primes $\lambda$ of $E$ with underlying rational prime $\ell$, the reduction satisfies both

$$\overline{\rho}_{\Pi,\lambda}\big|_{\operatorname{Gal}(\overline F/F(\mu_\ell)(\sqrt{\mathfrak d}))}$$

is absolutely irreducible and $\overline{\rho}_{\Pi,\lambda}$ is rigid for $(\mathcal P,\nu)$ in the sense of Definition~. This predicts simultaneous residual absolute irreducibility after the indicated cyclotomic-quadratic restriction and rigidity for almost all coefficient primes under the cuspidal base-change hypothesis.