Flat-rank one subgroups in hyperbolic totally disconnected groups

Suppose that $G$ is a hyperbolic, totally disconnected, locally compact group that does not fix a point of its hyperbolic boundary. A flat subgroup of flat-rank $1$ is a subgroup $H$ of $G$ with the stated flat-rank property, and its limit set is its limit set in the hyperbolic boundary. The subgroup $H_1$ denotes the subgroup associated with $H$ in the flat-subgroup structure, and an elliptic element is an element acting elliptically.

Flat-rank one subgroup conjecture. Every flat subgroup $H$ of flat-rank $1$ in $G$ has a limit set containing two elements, both fixed by $H$. Moreover, $H_1$ is relatively compact and is equal to the set of elliptic elements of $H$.

This conjecture asks whether the relationship between flat subgroups and the geometry of rough Cayley graphs observed for automorphism groups of trees and related settings also holds for hyperbolic totally disconnected, locally compact groups in general. The supplied text gives no resolution.