Large-data local BV scattering conjecture for the Einstein-scalar field system

Let initial data for the spherically symmetric Einstein-scalar field system have a BV norm, and let its maximal global development be the corresponding maximal globally defined spacetime solution. The development scatters locally in the BV norm if it has the assumed local BV scattering behavior in the region where the area of the $SO(3)$ symmetry orbits remains uniformly bounded.

Large-data local BV scattering conjecture. There exists initial data of arbitrarily large BV norm whose maximal global development scatters locally in the BV norm.

The conjecture would show that the locally BV-scattering class studied in the paper contains solutions arising from large initial data, extending the small-BV setting in which local BV scattering follows from Christodoulou's work. The source provides no resolution of this existence question.