Density conjecture for geometric-origin representations

Let $G$ be a linear algebraic group over the complex numbers, let $X$ be a normal complex variety with base point $x$, and write

$$\pi=\pi_1(X(\mathbb C),x),\qquad {\rm Ch}_{G,\mathbb C}^{\Box}(\pi)$$

for the framed character variety of $G$-representations of $\pi$. Fix an embedding $G\hookrightarrow \operatorname{GL}_{r,\mathbb C}$. A $G$-representation is of geometric origin if, after restriction to a suitable dense open subvariety and semisimplification, its associated linear local system is a direct summand of a local system arising from the higher direct images of a smooth projective morphism.

Density conjecture. The set of $G$-representations of geometric origin is Zariski dense in

$${\rm Ch}_{G,\mathbb C}^{\Box}(\pi).$$

This conjecture concerns the density of representations arising from geometry in framed character varieties. The source notes that it holds when $G$ is abelian, by the torus case of Esnault--Kerz, while Landesman and Litt showed that it is false in this generality because of rank obstructions for geometric local systems with infinite monodromy on certain curves.