Simpson's integrality conjecture for infinitesimally rigid representations

Let $M$ be a projective variety and let $\rho:\pi_1(M)\rightarrow\operatorname{SL}(n,\mathbb{C})$ be an infinitesimally rigid representation, meaning that its first cohomology vanishes. Then $\rho(\pi_1(M))$ is integral: there is a number field $k$ such that

$$\rho(\pi_1(M))\subseteq\operatorname{SL}(n,\mathcal{O}_k).$$

Simpson's conjecture. Every such representation has integral image. This connects rigidity of local systems on projective varieties with arithmeticity; the source presents it as Simpson's main conjecture in the projective case, and its general status is not specified here.