Link between trivial gauge transformations and power law weak fall-offs

Let $(M,\boldsymbol{g})$ be a $D$-dimensional Lorentzian manifold with conformal boundary $\partial M_c$. Let $(J, M, G)$ be an abelian gauge theory with gauge group $G$ and field boundary conditions $J|_{\partial M_c}$. Say that these boundary conditions are power law weak when the corresponding asymptotic charge at $\partial M_c$ is power law divergent. Let $(\widetilde{J}, M, G)$ denote the dual gauge theory. Link between trivial gauge transformations and power law weak fall-offs. The dual gauge theory has trivial gauge transformations at the conformal boundary $\partial M_c$ if and only if $J|_{\partial M_c}$ are power law weak. This proposes a general relation between duality and asymptotic boundary behaviour in abelian gauge theories, extending the preceding observation for dual $p$-form theories with Coulomb fall-offs. The parser supplies no evidence resolving the claim, so its status remains open.