Conjecture on polyhomogeneity for scattering solutions with infinitely regular data

Let $N\in\mathbb N_{\geq1}$, let $\phi$ solve $\Box_{\mathbb R^{2N+1,1}}\phi=0$ with no incoming radiation, and let $\psi^{\underline{\mathcal C}_{v_0}}\in H_{\mathrm b}^{a_0;\infty}(\underline{\mathcal C}_{v_0})$ with $a_0\geq -1/2$. For an outgoing null cone $\mathcal C_u$ with $u\leq u_0$, write $\mathcal A_{\mathrm{phg}}^{\overline{(0,0)}}(\mathcal C_u)$ for the relevant polyhomogeneous space. Polyhomogeneity conjecture. The restriction of the solution to every such cone satisfies

$$\psi|_{\mathcal C_u}\in \mathcal A_{\mathrm{phg}}^{\overline{(0,0)}}(\mathcal C_u).$$

This concerns the unresolved case of infinitely regular scattering data without a polyhomogeneous expansion. Finite angular regularity can fail to give summable higher-order coefficients, whereas polyhomogeneous data of infinite regularity are known to yield conformal smoothness; a related weaker question asks whether the radiation field has a finite limit at future null infinity.