Simpson's motivicity conjecture for rigid integrable connections

Let $X$ be a smooth complex projective variety. An integrable connection on $X$ is called rigid if it corresponds to an isolated point of the moduli space of integrable connections; it is motivic if it is a direct summand of a Gauss–Manin connection arising from a smooth projective family over a dense open subvariety of $X$.

Simpson's motivicity conjecture. Rigid integrable connections are motivic.

This is a weaker consequence of Simpson's standard conjecture when $X$ is defined over $\overline{\mathbb{Q}}$, because rigid connections correspond to zero-dimensional components of the de Rham moduli space. The conjecture remains open in the generality considered here.