Generic Schwarzschild Arnold-diffusion conjecture near the photon sphere

Let $N_\varepsilon$ denote the perturbed normally hyperbolic invariant manifold associated with the photon sphere, and let $L$ and $L_z$ be the angular-momentum quantities used in the Schwarzschild dynamics. Generic Schwarzschild diffusion conjecture near the photon sphere. For a generic stationary perturbation of the Schwarzschild 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

$$|L_z(T)-L(0)|<\delta,\qquad |L_z(T')+L(0)|<\delta.$$

For a generic perturbation that is $1$-periodic in $\tau$, for every $\delta>0$ there is an orbit visiting every $\delta$-ball centered on $N\times\mathbb T^1\times(4/9,1/2)$, with the center written $(\Xi,\tau,E)$, provided $\varepsilon$ is sufficiently small. These are explicitly presented as part of a conjectural picture based on normal hyperbolicity and the scattering mechanism.