Non-negativity conjecture for the mass functional of GB initial data

Let $(\rho,z)$ be coordinates on an orbit space $\mathcal{B}$, and let $(v,\lambda',Y)$ denote the variables defining a GB initial data set. The mass functional is denoted by $\mathcal{M}(v,\lambda',Y)$.

Non-negativity conjecture. Consider a GB initial data set. Then $\mathcal{M}\left(v,\lambda',Y\right)$ is a non-negative functional for any orbit space.

Positivity of $\mathcal{M}$ is known only for orbit spaces $\mathcal{B}\in\Xi$; the conjecture asserts non-negativity for all orbit spaces on asymptotically flat initial data sets with positive scalar curvature.