  Does every one-ended $CAT(0)$ group have semistable fundamental group at
infinity? As we write, this is an open question. Let $G$ be such a group acting
geometrically on the proper $CAT(0)$ space $X$. In this paper we show that in
order to establish a positive answer to the question it is only necessary to
check that any two geodesic rays in $X$ are properly homotopic. We then show
that if the answer to the question is negative, with $(G,X)$ a counter-example,
then the boundary of $X$, $\del X$ with the cone topology, must have a weak cut
point. This is of interest because a theorem of Papasoglu and the second-named
author \cite{PS} has established that there cannot be an example of $(G,X)$
where $\del X$ has a cut point. Thus, the search for a negative answer comes
down to the difference between cut points and weak cut points. We also show
that the Tits ball of radius $\frac{\pi}{2}$ about that weak cut point is a
"cut set" in the sense that it separates $\del X$. Finally, we observe that if
a negative example $(G, X)$ exists then $G$ is rank 1.
