The horofunction Morse characterization via ideal triangle points

Let $X$ be a proper complete CAT(0) space, let $c$ be a geodesic ray starting at $p$, and let $h=b_c$ be the corresponding horofunction. For each $x\in X$, let $(i_{\infty})_x$ denote the distinguished point associated to the triangle based on $x,p,c$.

Morse characterization. The horofunction $h=b_c$ is Morse if and only if there exists a $\delta$ such that, for every $x\in X$, $(i_{\infty})_x$ lies in the $\delta$-neighborhood of $\operatorname{im}(c)$.

This gives a geometric characterization of Morse horofunctions in terms of the ideal points of triangles based at infinity. The source does not provide evidence resolving the conjecture.