Generic Kerr bounded-phase-space diffusion conjecture

Define the bounded-motion region

$$\mathcal B(C):=\left\{0<E^2<1,\ |L_z|<C,\ 0<Q<C\mid -R(r^*)>0,\ -R(r_*)<0\right\}\cap H^{-1}\left(-\frac12\right),$$

where $R$ is the Kerr radial potential, $r^*$ is the local maximum of $-R$, and $r_*$ is the local minimum of $-R$ to its right. Generic Kerr bounded-phase-space diffusion conjecture. For a generic stationary perturbation of the Kerr metric, for every $\delta>0$ and $C>0$, an orbit exists that is $\delta$-dense on $\mathcal B(C)$, provided $\varepsilon$ is small. The same assertion holds for a generic perturbation that is $1$-periodic in $\tau$.

This is the Kerr analogue of the conjectured dense-orbit statement for bounded Schwarzschild motions. It is motivated by applying Arnold diffusion to the Liouville-Arnold region, but the source gives no resolution evidence.