The compatible Galois representations conjecture for regular L-algebraic cuspidal representations

Let $L$) be a number field and let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A_L)$ that is $L$-algebraic and regular. For a prime $p$ and an isomorphism $\iota:\overline{\mathbb Q}_p\cong\mathbb C$, write $\Gamma_L$ for the absolute Galois group of $L$. The compatible Galois representations conjecture. There exists a continuous, semisimple representation

$$r_{p,\iota}(\pi):\Gamma_L\to\operatorname{GL}_n(\overline{\mathbb Q}_p)$$

such that, for all but finitely many finite places $v$ of $L$ at which $\pi_v$ is unramified, the restriction $r_{p,\iota}(\pi)|_{\Gamma_{L_v}}$ is unramified and the semisimple conjugacy class of $r_{p,\iota}(\pi)(\operatorname{Frob}_v)$ equals the Satake parameter of $\iota^{-1}\pi_v$. This predicts the Galois representations attached to regular $L$-algebraic automorphic representations of general linear groups; the statement is presented as a conjecture here, with no resolution specified in the source.