Nonexistence conjecture for nonpositive mass-aspect types

Let $\rho$ be the energy density and $J$ the momentum density on a smooth, conformally compact, geometrically finite general relativistic initial data set satisfying the energy condition

$$\rho \geq |J|.$$

A mass aspect function is said to be of ${\cal H}_n$-type, ${\cal P}_n^q$-type, or ${\cal E}_{m_{\mathrm c}}$-type according to the corresponding asymptotic classification, with $n\geq 1$ and $m_{\mathrm c}<-1$. Nonexistence conjecture. Mass aspect functions of ${\cal H}_n$- and ${\cal P}_n^q$-type with $n\geq 1$, and of ${\cal E}_{m_{\mathrm c}}$-type with $m_{\mathrm c}<-1$, do not arise on such initial data sets. The conjecture would provide a related positive energy result in $(2+1)$ dimensions, where whether a general positive energy theorem holds remains unresolved.