Kehle's boundedness conjecture for Lebesgue-generic Kerr--AdS black holes

Let $\mathcal P$ be the parameter space of dimensionless Kerr--AdS black hole parameters $(\mathfrak m,\mathfrak a):=(M\sqrt{-\Lambda},a\sqrt{-\Lambda})$. Let $\mathcal P_{\mathrm{Bounded}}\subset\mathcal P$ be the set of parameters under consideration, and let $\psi$ solve the wave equation with $\alpha=2$. Let $\mathcal{CH}$ denote the Cauchy horizon. Kehle's boundedness conjecture. The linear $C^0$ conjecture is false for Lebesgue-generic Kerr--AdS black holes: for parameters in $\mathcal P_{\mathrm{Bounded}}$, the perturbation remains uniformly bounded and extends continuously across the Cauchy horizon,

$$|\psi(x)|\le C,$$

where $\mathcal P_{\mathrm{Bounded}}$ is dense, has full Lebesgue measure, and is Baire-exceptional.

This conjecture contrasts Lebesgue-genericity with Baire-genericity: the paper explains that the $C^0$ formulation is expected to be true for Baire-generic Kerr--AdS parameters but false for Lebesgue-generic parameters. The source attributes the conjecture to Kehle (2020).