The polynomial decay conjecture for the linearised equation

Let $\psi^{\mathrm{dat}}$ be data as in the paper's scattering-data definition, with $d>0$ and $N>5$, for the linearised equation on the hypersurface $\Sigma_1$. Let $\Theta^{\mathrm{mom}}$, $\Theta^{\mathrm{com}}$, and $\Theta^\Lambda$ denote the corresponding obstruction quantities, and let $J_+$ be the unstable-mode detection current. Polynomial decay conjecture. If

$$\Theta^{\mathrm{mom}}[\psi]=\Theta^{\mathrm{com}}[\psi]=\Theta^\Lambda[\psi]=\Sigma_1[J_+]=0,$$

then the forward solution to the linearised equation with $\psi^{\mathrm{dat}}$ as initial data decays polynomially as $t\to\infty$. The vanishing conditions are intended to remove the obstructions to boundedness and decay identified through the projection operators; the paper presents polynomial decay under these conditions as a conjecture.