Density conjecture for pleated surfaces in the complex character variety

Let $S$ be a closed surface with fundamental group $\Gamma$, let $d\geq 2$, and let $\lambda$ be a maximal geodesic lamination on $S$. Denote by $\mathfrak R(\lambda,d)$ the space of conjugacy classes of $d$-pleated surfaces with pleating locus $\lambda$, and by $\mathfrak X(\Gamma,\operatorname{\mathsf{PGL}}_d(\operatorname{\mathbb{C}}))$ the character variety of representations of $\Gamma$ into $\operatorname{\mathsf{PGL}}_d(\operatorname{\mathbb{C}})$. Density conjecture. For every maximal geodesic lamination $\lambda$, the space $\mathfrak R(\lambda,d)$ is dense in

$$\mathfrak X(\Gamma,\operatorname{\mathsf{PGL}}_d(\operatorname{\mathbb{C}})).$$

Equivalently, every representation from $\Gamma$ to $\operatorname{\mathsf{PGL}}_d(\operatorname{\mathbb{C}})$ can be approximated arbitrarily well by generalized bending deformations along $\lambda$ of Hitchin representations. The preceding results show that these deformations meet every connected component of the character variety, but density within each component remains open.