  In this paper, we expand on results from our previous paper "The Case Against
Smooth Null Infinity I: Heuristics and Counter-Examples" [1] by showing that
the failure of "peeling" (and, thus, of smooth null infinity) in a
neighbourhood of $i^0$ derived therein translates into logarithmic corrections
at leading order to the well-known Price's law asymptotics near $i^+$. This
suggests that the non-smoothness of $\mathcal{I}^+$ is physically measurable.
More precisely, we consider the linear wave equation $\Box_g \phi=0$ on a fixed
Schwarzschild background ($M>0$), and we show the following: If one imposes
conformally smooth initial data on an ingoing null hypersurface (extending to
$\mathcal{H}^+$ and terminating at $\mathcal{I}^-$) and vanishing data on
$\mathcal{I}^-$ (this is the no incoming radiation condition), then the precise
leading-order asymptotics of the solution $\phi$ are given by
$r\phi|_{\mathcal{I}^+}=C u^{-2}\log u+\mathcal{O}(u^{-2})$ along future null
infinity, $\phi|_{r=R>2M}=2C\tau^{-3}\log\tau+\mathcal{O}(\tau^{-3})$ along
hypersurfaces of constant $r$, and $\phi|_{\mathcal{H}^+}=2Cv^{-3}\log
v+\mathcal{O}(v^{-3})$ along the event horizon. Moreover, the constant $C$ is
given by $C=4M I_0^{(\mathrm{past})}[\phi]$, where
$I_0^{(\mathrm{past})}[\phi]:=\lim_{u\to -\infty}
r^2\partial_u(r\phi_{\ell=0})$ is the past Newman--Penrose constant of $\phi$
on $\mathcal{I}^-$. Thus, the precise late-time asymptotics of $\phi$ are
completely determined by the early-time behaviour of the spherically symmetric
part of $\phi$ near $\mathcal{I}^-$. Similar results are obtained for
polynomially decaying timelike boundary data. The paper uses methods developed
by Angelopoulos--Aretakis--Gajic and is essentially self-contained.
