Weak Cederbaum–Sakovich conjecture on compatible asymptotic coordinates

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 $z^iR_g\in L^1(M,\Phi_z)$. A coordinate chart $\{\bar z^i\}_{i=1}^3$ is called compatible when its coordinate change from $\{z^i\}_{i=1}^3$ is asymptotic to the identity. Weak Cederbaum–Sakovich conjecture. There is a geometric condition on the coordinates $\{z^i\}_{i=1}^3$ ensuring the existence of a compatible asymptotic chart $\{\bar z^i\}_{i=1}^3$ such that the center of mass converges in the $\bar z$-coordinates. This is proposed as a weaker reformulation of the Cederbaum–Sakovich conjecture, motivated by examples where the center of mass fails to converge in a specified asymptotic chart; the source gives no resolution.