More precise asymptotics of quasinormal modes near extremality

Let $\sigma_0$ be a simple quasinormal mode of the Klein--Gordon equation on the near-horizon geometry with parameters $r_{\rm e}$ and $r_{\rm c}$. Let $\epsilon>0$ be small and set $r_{\rm C}=r_{\rm e}-2\epsilon$. More precise asymptotics of quasinormal modes. The unique nearby quasinormal mode on RNdS depends on $\epsilon\in[0,r_{\rm e}/2)$ in a smooth or polyhomogeneous fashion. This is posed as a problem for future work concerning how near-horizon quasinormal modes lift to quasinormal modes of a near-extremal Reissner--Nordström--de Sitter spacetime; the supplied text does not state whether the claim has been proved.