The integrality consequence for rigid local systems

Let $X$ be a smooth projective variety, and let $L$ be a rigid local system on $X$. A local system is integral when the traces of its monodromy matrices are algebraic integers. The integrality consequence. Any rigid local system $L$ on $X$ is integral. The source presents this as a consequence of the geometric-origin conjecture, since local systems of geometric origin inherit integral structure. It also notes that Esnault and Groechenig had posted a proof for cohomologically rigid local systems, but the general statement is not resolved in the supplied text.