The non-quasiconvex subgroup conjecture for Cannon–Thurston maps

Let $H<G$ be a hyperbolic subgroup of a hyperbolic group $G$. Suppose the Cannon–Thurston map

$$\partial_{H,G}:\partial_{\infty}H\to\partial_{\infty}G$$

exists, and suppose that $H$ is not quasiconvex in $G$. Non-quasiconvex subgroup conjecture. There is a continuum of nonconical limit points of $H$ in $G$ whose preimages under $\partial_{H,G}$ are not singletons. This conjecture is motivated by the fact that conical limit points have singleton preimages under the Cannon–Thurston map; it predicts that failure of quasiconvexity produces continuum many nonconical limit points with nonsingleton fibers.