  A convenient approach to analyze spatial infinity is to use a cylinder
representation $I$ and impose a gauge based on a congruence of conformal
geodesics. This so-called conformal Gauss gauge comes along with the freedom to
specify initial data for the conformal geodesics. Such a gauge has been
constructed from an ordinary Cauchy surface and from past null infinity
$\mathcal{J}^-$, respectively. The purpose of this note is to compare these
gauges near the critical set $I^-$, where $I$ "touches" $\mathcal{J}^-$, as it
turns out that they are related in a somewhat unexpected intricate way.
