Simpson's geometricity conjecture for quasi-geometric flat bundles

Let $Y$ be the complement of a normal crossing divisor in a smooth proper Deligne–Mumford stack. A flat bundle on $Y$ is quasi-geometric if it extends as an algebraic regular-singular flat connection over the compactification and its monodromy representation is conjugate to one with coefficients in $\overline{\mathbb{Q}}$. A flat bundle is geometric if it is a subquotient of a Gauss–Manin flat bundle arising from a smooth proper family over a dense open subset. Simpson's conjecture. Any semisimple quasi-geometric flat bundle on $Y$ is geometric. This is a partial converse to the fact that Gauss–Manin connections are quasi-geometric. The source records that the relevant existence statement for such Hodge structures has since been proved, so this conjecture is treated as solved.