Automorphic-to-Galois compatibility conjecture for GL(n)

Let $H$ be a multiset of $n$ integers and let $\pi$ be an irreducible constituent of the cuspidal automorphic spectrum with infinitesimal character $H$. After identifying $\overline{\mathbb{Q}}\subset\mathbb{C}$, each local Langlands parameter $\operatorname{rec}_p(\pi_p)$ should be definable over $\overline{\mathbb{Q}}$.

Automorphic-to-Galois compatibility conjecture. There is an irreducible geometric strongly compatible system $\mathcal{R}$ of l-adic representations such that

$$\operatorname{HT}(\mathcal{R})=H,\qquad \operatorname{WD}_p(\mathcal{R})^{\operatorname{ss}}=\operatorname{rec}_p(\pi_p)\quad\text{for all primes }p.$$

This predicts the Galois representation attached to a cuspidal automorphic representation and compatibility at every finite place. The source gives no general resolution status.