Generic Schwarzschild diffusion conjecture on bounded phase space

Define the bounded-motion region

$$\mathcal B(C):=\left\{ V(r_+)<\frac{E^2}{2}<\min\{V(r_-),V(\infty)\},\quad |L_{isco}|<|L|<C\right\}\cap H^{-1}\left(-\frac12\right).$$

Here $E$ is the particle energy, $L$ its angular momentum, $V$ the effective potential, and $H$ the Hamiltonian. Generic Schwarzschild bounded-phase-space diffusion conjecture. For a generic stationary perturbation of the Schwarzschild metric, for every $C>0$ and $\delta>0$, 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 $\tau$.

The claim is the bounded-motion analogue of Arnold's diffusion conjecture, formulated separately for stationary and time-periodic perturbations. The source presents both assertions as conjectural and gives no resolution evidence.