Fontaine–Mazur–Petrov conjecture for ℓ-adic local systems

Establish that an irreducible ℓ-adic local system \mathbb{V} on X_{K^s} is of geometric origin if and only if its isomorphism class has finite orbit under \operatorname{Gal}(K^s/K).

Background

This conjecture is the non-abelian ℓ-adic analogue of the Tate conjecture. It predicts that ℓ-adic local systems of geometric origin are precisely those with finite Galois orbits after descending from an algebraically closed field. It is known in certain cases (e.g., curves over finite fields via Lafforgue).