The lexicographic free-group conjecture for finitely presented \Lambda-free groups

Let $\Lambda$ be an ordered abelian group. A group is $\Lambda$-free if it acts freely by isometries on a $\Lambda$-tree. Let $G$ be a finitely presented $\Lambda$-free group.

Lexicographic free-group conjecture. $G$ is $\mathbb{Z}^k$-free for an appropriate $k\in\mathbb{N}$, where $\mathbb{Z}^k$ is ordered lexicographically.

The conjecture predicts that every finitely presented group acting freely on a $\Lambda$-tree can be represented by a free action on a lexicographically ordered finitely generated free abelian group. The paper establishes the analogous statement with $\mathbb{R}^n$ in place of $\mathbb{Z}^k$, but the stated conjecture remains open.