Integrality conjecture for rigid Zariski-dense representations of property T groups

Let $X$ be a smooth projective variety, let $G$ be the Lie group under consideration, and let $\varrho:\pi_{1}(X)\longrightarrow G$ be a rigid Zariski-dense representation. Assume that $\pi_{1}(X)$ has Kazhdan's property $T$.

Property $T$ integrality conjecture. The representation $\varrho$ is integral.

The paper describes this as a more approachable special case after noting that the full versions of its preceding conjectures seem out of reach. No resolution is given in the supplied text.