The global Azumaya algebra conjecture for automorphic Galois representations

Let $K$ be a number field, let $\pi$ be a regular cuspidal algebraic automorphic representation of $\operatorname{GL}_n({\mathbb A}_K)$, let $\rho_{\pi,p}$ be its associated Galois representation, and let $D_p$ be the Azumaya algebra associated to it.

Global Azumaya algebra conjecture. There exists a central simple algebra $D$ over $\mathbb Q(\pi)$ such that, for every prime $p$,

$$D_p\simeq D\otimes_{\mathbb Q(\pi)}\mathbb Q(\pi)_p.$$

This predicts that the local Azumaya algebras arising from the Galois representations are the completions of one global central simple algebra. The source motivates it through the expected Mumford–Tate description and cites Chenevier for the construction of $D_p$; no resolution is given.