Generic Kerr Arnold-diffusion conjecture near the photon shell

Let $N_\varepsilon$ be the perturbed normally hyperbolic invariant manifold associated with the Kerr photon shell, and let $(a,M,E,Q,L_z)$ satisfy the conditions in assumption (2) of Theorem 1. For the relevant interval $\ell(\mathcal R)$, let $\ell_-(E)<\ell_+(E)$ be its two endpoints. Generic Kerr diffusion conjecture near the photon shell. For a generic stationary perturbation of the Kerr metric, for every $\delta>0$ there is an $\varepsilon_0$ such that, whenever $|\varepsilon|<\varepsilon_0$, an orbit on $N_\varepsilon$ and times $T,T'$ exist with

$$|L_z(T)-E\ell_-(E)|<\delta,\qquad |L_z(T')-E\ell_+(E)|<\delta.$$

For a generic perturbation that is $1$-periodic in $\tau$, for every $\delta>0$ there is an orbit visiting every $\delta$-ball centered at $\{H_{\hat N_\varepsilon}=-1/2\}$ on $\hat N_\varepsilon$, provided $\varepsilon$ is sufficiently small. These assertions extend the Schwarzschild diffusion picture to Kerr, but the source supplies no resolution evidence.