Cannon–Thurston conjecture for surface Kleinian groups

Suppose a surface group $\pi_1(S)$ acts freely and properly discontinuously on $\mathbb{H}^3$ by isometries. Then the inclusion

$$\tilde{i}: \widetilde{S^h}\rightarrow \mathbb{H}^3$$

extends continuously to the boundary.

Cannon–Thurston conjecture. The inclusion $\tilde{i}:\widetilde{S^h}\rightarrow\mathbb{H}^3$ extends continuously to the boundary.

For a simply degenerate group, this is equivalent to asking whether the limit set is locally connected. The conjecture is presented as a proposal of Cannon and Thurston for surface Kleinian groups.