Density of mapping class group orbits for non-elementary non-discrete representations

Let $\Gamma_o$ be the group of the six-point configuration and let $X_0$ denote the relevant character space of representations into $\mathrm{PSL}_2(\mathbb{R})$. For a representation $\rho\colon\Gamma_o\rightarrow\mathrm{PSL}_2(\mathbb{R})$, call it non-elementary if its image is non-elementary and non-discrete if its image is not a discrete subgroup. The braid group $B_6(S^2)$ acts on the character class $[\rho]$. Density conjecture. For every non-elementary, non-discrete representation $\rho\colon\Gamma_o\rightarrow\mathrm{PSL}_2(\mathbb{R})$, the orbit

$$B_6(S^2)\cdot[\rho]$$

is dense in $X_0$. This would extend the expected ergodicity phenomenon beyond the representation spaces treated by the main theorem, but the source presents it only as a reasonable statement and gives no resolution.