Conjecture on directions and the hyperbolic boundary of totally disconnected groups

Let $G$ be a hyperbolic, totally disconnected, locally compact group. An element has non-trivial scale when its scale is not equal to $1$, and its direction is the corresponding equivalence class used to define the set of directions of $G$. The pseudo-distance is the pseudo-metric on directions associated with this structure.

Directions and boundary conjecture. The map that assigns to each element of non-trivial scale its attracting boundary point defines an injection from the set of directions of $G$ into the hyperbolic boundary. Elements of $G$ with distinct directions have pseudo-distance $2$.

The conjecture seeks a common extension of the authors' theorem and a result concerning directions in totally disconnected, locally compact groups. Its status is not established in the supplied text.