Shallow quasinormal-mode convergence conjecture

Fix $0<r_{\rm e}<r_{\rm c}$. Let $\varsigma$ range over the quasinormal modes $\operatorname{QNM}(r_{\rm C},r_{\rm e},r_{\rm c},mu)$ of the Klein–Gordon equation, let $\kappa_{\rm C}$ denote the cosmological-horizon surface gravity, and let $C_0$ be the constant from Theorem I. Let $\operatorname{QNM}_{\rm NH}(mu)$ denote the quasinormal modes of the near-horizon geometry.

Shallow-QNM convergence conjecture. As $r_{\rm C} earrow r_{\rm e}$, the set

$$\left\{\frac{\varsigma}{\kappa_{\rm C}}:\varsigma\in\operatorname{QNM}(r_{\rm C},r_{\rm e},r_{\rm c},\mu),\ \operatorname{Im}\varsigma>-C_0\kappa_{\rm C}\right\}$$

converges to

$$\operatorname{QNM}_{\rm NH}(\mu)\cap\{\operatorname{Im}\sigma>-C_0\}.$$

This predicts that all sufficiently shallow QNMs, after rescaling by the cosmological surface gravity, are captured by the near-horizon spectrum in the near-extremal limit. The statement is posed as a future problem and is not resolved in the paper.