Full polyhomogeneity of the spacetime metric

Let $(\gamma,k)$ be initial data as in Theorem~, let $\mathcal E_0$ be an index set with $\min\operatorname{Re}\mathcal E_0>1$, and suppose

$$\gamma-\gamma_{b_0}\in\mathcal A^{\mathcal E_0}(\Sigma_{\mathrm{IVP}}),\qquad k-k_{b_0}\in\mathcal A^{\mathcal E_0+1}(\Sigma_{\mathrm{IVP}}).$$

Let $g$ be the spacetime metric produced by Theorem~, and let $h$ denote its perturbation on $\phi_\mathsf{S}(\Omega)$. Full polyhomogeneity of the spacetime metric. The metric $g$ is polyhomogeneous on $\phi_\mathsf{S}(\Omega)$, with

$$h\in\mathcal A^{\mathcal E_0,\,\langle\mathcal E_{\!\mathscr I}^{\mathcal C}\rangle,\,\mathcal E_+,\,\mathcal E_\mathcal K}(\phi_\mathsf{S}(\Omega)),$$

where $\mathcal E_\mathcal K$ is an index set satisfying $\min\operatorname{Re}\mathcal E_\mathcal K\geq 3$ and

$$\min\operatorname{Re}(\mathcal E_\mathcal K\setminus\{(3,0)\})>3.$$

This would strengthen the asymptotic information supplied by Theorem~ from the stated conormal regularity to full polyhomogeneous expansions. The claim is presented as a future avenue and no proof or resolution is given.