Existence conjecture for singular Kerr waves from compactly supported data

Let $\Sigma_0$ be a complete $2$-ended asymptotically flat Cauchy hypersurface in a Kerr spacetime with $0<|a|<M$, and let $\psi$ solve the linear wave equation

$$\Box_g\psi=0.$$

The relevant assumptions are the polynomial upper and lower bounds on the energy along the event horizon specified in the paper's main theorem. Kerr event-horizon energy conjecture. There exist smooth and compactly supported initial data on $\Sigma_0$ such that the corresponding solutions obey those energy assumptions along the event horizon. This conjecture would, together with the paper's main theorem, yield singular solutions arising from smooth compactly supported initial data; linearity would then give genericity. The corresponding existence and instability result is known in the Reissner–Nordström setting, but remains open for Kerr.