Hod–Piran charged massless scalar decay conjecture

Let $\mathcal{H}^+$ be the event horizon, let $v$ be the standard advanced time null coordinate, let $e$ be the asymptotic charge at timelike infinity, and define

$$\delta(q_0e)=1-\operatorname{Re}\sqrt{1-4(q_0e)^2}\in[0,1).$$

Charged massless scalar decay conjecture. For smooth, regular, generic admissible data with a non-empty black hole and $m^2=0$,

$$|\phi|_{\mathcal{H}^+}(v)\sim v^{-2+\delta(q_0e)},\qquad |D_v\phi|_{\mathcal{H}^+}(v)\sim v^{-2+\delta(q_0e)}.$$

The conjecture is supported by heuristic and numerical work; an upper bound is proved on a fixed Reissner–Nordström background for small charge, but the full nonlinear asymptotic remains open.