Linear conjugator conjecture for Morse elements

Let $G$ be a finitely generated group. An element of $G$ is Morse if its orbit in a Cayley graph of $G$ is a quasi-geodesic such that every $(K,C)$-quasigeodesic with endpoints on this orbit remains in a uniformly bounded neighborhood of it. The set of Morse elements is said to satisfy the linear conjugator property if there are constants $K,C$ such that whenever Morse elements $a,b\in G$ are conjugate, there exists $g\in G$ with

$$ga=bg$$

and

$$|g|\leq K(|a|+|b|)+C.$$

Linear conjugator conjecture. In every finitely generated group, the set of Morse elements satisfies the linear conjugator property.

The theorem preceding this conjecture establishes the property for Morse elements in hierarchically hyperbolic groups, while the conjecture asks for the same linear bound in all finitely generated groups. The general case is left open.