Integral-points conjecture for Betti moduli spaces

Let $X$ be a smooth projective complex variety, and let $M_B(X,r)$ denote its Betti moduli space of rank $r$ representations as defined in the paper. An integral point is a point defined over the ring $\overline{\mathbb{Z}}$ of algebraic integers. Integral-points conjecture. Each component of $M_B(X,r)$ has a $\overline{\mathbb{Z}}$-point. This arithmetic conjecture supplies the integral-point hypothesis needed to deduce finite monodromy for isomonodromy foliations; the source describes it as a slight extension of Simpson's integrality conjecture and does not claim a general proof.