Marked-polytope invariance conjecture for nice (2,1)-presentations

Let $G$ be a group admitting a nice $(2,1)$-presentation $\pi=\langle x,y\mid r\rangle$, and let $\mathcal{M}_\pi\subset H_1(G;\mathbb{R})$ be its marked polytope. Two subsets are translates when one is obtained from the other by adding an element of $H_1(G;\mathbb{R})$. Marked-polytope invariance conjecture. Up to translation, the marked polytope $\mathcal{M}_\pi$ is an invariant of $G$. The marked polytope is known to be unchanged up to translation under cyclic permutation of the relator, and the conjecture asserts that this is the only indeterminacy; the authors note that no satisfactory theory relates arbitrary $(2,1)$-presentations of the same group.