Massive scalar-field $v^{-5/6}$ tail conjecture

Let $\mathcal{H}^+$ be the event horizon and let $v$ be the standard advanced time null coordinate. Massive scalar-field tail conjecture. For smooth, regular, generic admissible data with a non-empty black hole, in the massive case $m^2\neq0$ and $q_0\in\mathbb{R}$,

$$|\phi|_{\mathcal{H}^+}(v)\sim |\sin(mv+o(v))|v^{-5/6},\qquad |D_v\phi|_{\mathcal{H}^+}(v)\sim |\sin(mv+o(v))|v^{-5/6}.$$

The claim is based on heuristic and numerical studies for uncharged massive fields and arguments for charged massive fields; the general nonlinear result remains open.