The strict compatibility and automorphy conjecture

Let $K$ be a number field, let $M$ be a coefficient field, and let $(r_\lambda)$ be a weakly compatible system of Galois representations of rank $n$. For each finite place $v$ of $K$, let $\mathrm{WD}_v$ denote the associated Weil–Deligne representation, and let $\operatorname{rec}_{K_v}$ be the local Langlands correspondence.

Strict compatibility and automorphy conjecture. Any weakly compatible system of Galois representations is strictly compatible, and is in addition automorphic: there is an algebraic automorphic representation $\pi$ of $\operatorname{GL}_n(\mathbb A_K)$ such that

$$\mathrm{WD}_v\cong\operatorname{rec}_{K_v}(\pi_v|\det|^{(1-n)/2})$$

for each finite place $v$ of $K$.

The conjecture combines the expected strict compatibility of weakly compatible systems with their automorphy and is the target of progress from automorphy lifting theorems. The source gives no resolution of the full statement.