Simpson's geometric origin conjecture for rigid local systems

Let $X$ be a smooth projective complex variety, and let $\mathcal{L}$ be an irreducible rigid local system on $X$, meaning that it cannot be deformed continuously or algebraically to a non-isomorphic local system. A local system may be viewed in the Betti or de Rham sense, which are equivalent over $\mathbb{C}$ by the Riemann–Hilbert correspondence. Simpson's conjecture. The local system $\mathcal{L}$ is of geometric origin: there exists an open dense subset $U\subset X$ and a smooth projective family

$$\pi:Y\to U$$

such that $\mathcal{L}$ is a direct summand of

$$R^{\bullet}\pi_*\operatorname{\mathcal{O}}_Y.$$

This conjecture predicts that rigid local systems arise from the cohomology of smooth projective families, linking isolated points in moduli spaces of local systems with geometric and arithmetic constructions. Its general status is not established here.