Escaping set rigidity for exponential maps with singular value in the Julia set

Let $E_{\kappa}(z)=e^z+\kappa$ be an exponential map, let $J(E_{\kappa})$ denote its Julia set, and let $I(E_{\kappa})$ denote its escaping set. An order-preserving conjugacy between two escaping sets is a conjugacy that preserves the natural order of the dynamic rays. Escaping set rigidity conjecture. Suppose that $\kappa_1$ is a parameter with $\kappa_1\in J(E_{\kappa_1})$, and let $\kappa_2\notin\\{\kappa_1+2\pi i k:k\in\mathbb{Z}\\}$. Then there exists no order-preserving conjugacy

$$f:I(E_{\kappa_1})\to I(E_{\kappa_2})$$

between $E_{\kappa_1}$ and $E_{\kappa_2}$. Rigidity of the escaping dynamics is expected to obstruct quasiconformal conjugacies between nearby non-hyperbolic stable parameters and would imply density of hyperbolicity. The source provides no resolution of this conjecture.