The non-linearity conjecture for Kac-Moody groups with hyperbolic Weyl group

Let $\Lambda$ be a Kac-Moody group over $\mathbf{F}_q$ with Weyl group $W$, and suppose that $W$ is Gromov-hyperbolic. The notation $q>\!\!>1$ means that $q$ is sufficiently large.

Non-linearity conjecture. If $W$ is Gromov-hyperbolic and $q>\!\!>1$, then $\Lambda$ is not linear.

This proposes that non-linearity holds for a much wider class of discrete Kac-Moody groups than the examples established in the paper. The source does not state a resolution of the conjecture.