Rigidity conjecture for type-preserving representations of 2-bridge links

Let \rho:\pi_1(\text{\boldmathT})\to PSL(2,\mathbb{C}) be a type-preserving representation such that its end-invariant set satisfies $\mathcal{E}(\rho)=\Lambda(\hat\Gamma_r)$. Here $\Lambda(\hat\Gamma_r)$ is the limit set associated with the hyperbolic $2$-bridge link parameter $r$, and $\rho_r$ is the representation induced by the holonomy representation of the hyperbolic $2$-bridge link $K(r)$. Rigidity conjecture. The representation $\rho$ is conjugate to $\rho_r$. This is presented as a variation of a special case of a question of Bowditch and a conjecture of Tan, Wong and Zhang; no resolution is given in the source.