Modularity conjecture for geometric Galois representations

Let $F$ be a number field, let $\ell$ be a prime, and let $\rho:\operatorname{Gal}(\overline{F}/F)\to\operatorname{GL}_n(\overline{\mathbb{Q}}_\ell)$ be a Galois representation that comes from geometry, meaning it is unramified outside finitely many places and occurs as a subquotient of a finite direct sum of étale cohomology groups of smooth projective varieties, with Tate twists. Fix an isomorphism $\iota:\overline{\mathbb{Q}}_\ell\simeq\mathbb{C}$. Modularity conjecture. There exists an isobaric automorphic representation $\Pi$ of $\operatorname{GL}_n(\mathbb{A}_F)$ such that

$$L_v(s,\Pi)=L_v(s,\rho)$$

at every finite place $v$ and at $v=\infty$, hence $L(s,\Pi)=L(s,\rho)$ and $\Lambda(s,\Pi)=\Lambda(s,\rho)$. This is the automorphic form of the expected reciprocity law between geometric Galois representations and automorphic representations; no general resolution is stated.