Density conjecture for geometric-origin representations on special subschemes

Let $X$ be a normal complex variety, let $x\in X(\mathbb C)$, and let $\pi=\pi_1(X(\mathbb C),x)$. Let

$$Z\hookrightarrow {\rm Ch}_{\operatorname{GL}_r,\mathbb C}(\pi)$$

be a special subscheme. A point of $Z$ is of geometric origin when it corresponds to a representation $\rho\colon \pi\to \operatorname{GL}_r(\mathbb C)$ whose associated local system is of geometric origin.

Special-locus density conjecture. The set of complex points of $Z$ corresponding to representations of geometric origin is dense in $Z$. In particular, the special points are dense on $Z$.

This extends the paper's density question from the full character variety to special, or arithmetic, subschemes. The source gives no resolution of this special-locus conjecture; it refers back to the obstruction showing that the unrestricted density conjecture is false in general.