Cederbaum–Sakovich conjecture on convergence of the Riemannian center of mass

Let $(M^3,g)$ be an asymptotically Euclidean manifold of order $\tau\in(-1,-\frac{1}{2})$ with respect to a structure of infinity $\Phi_z$ and coordinates $\{z^i\}_{i=1}^3$. Assume that $z^iR_g\in L^1(M,\Phi_z)$, where $R_g$ is the scalar curvature, and let the center-of-mass expression referred to in the source be defined in the $z$-coordinates. Cederbaum–Sakovich conjecture. There is a geometric condition on the coordinates $\{z^i\}_{i=1}^3$ ensuring that the center of mass converges in the $z$-coordinates. This is the Riemannian, time-symmetric version of the conjecture concerning convergence of the center of mass for general relativistic initial data; the source does not state whether it has been resolved.