Non-spherical stability conjecture for locally BV-scattering Einstein-scalar field solutions

Consider a spherically symmetric dispersive solution of the Einstein-scalar field equations that scatters locally in the BV norm. A perturbation is non-spherically symmetric if it does not preserve spherical symmetry under the $SO(3)$ action.

Non-spherical stability conjecture. Spherically symmetric locally BV-scattering dispersive solutions to the Einstein-scalar field equations are stable against non-spherically symmetric perturbations.

The corresponding stability under small, regular, spherically symmetric perturbations follows from the decay result in the paper. Stability under non-spherically symmetric perturbations is presented as a more ambitious conjecture, and the source gives no resolution.