Friedl–Tillmann invariance conjecture for marked polytopes

Let $G$ be a group admitting a nice $(2,1)$-presentation $f\pi=\langle x,y\mid r\rangle$, and let $\mathcal{M}_\pi\subset H_1(G;\mathbb{R})$ be its associated marked polytope. Friedl–Tillmann invariance conjecture. The marked polytope $\mathcal{M}_\pi$ is an invariant of $G$ up to translation. More formally, if $f\colon G_\pi\to G_{\pi'}$ is an isomorphism of groups associated to $(2,1)$-presentations $\pi$ and $\pi'$, then the associated polytopes satisfy

$$\mathcal{P}_{\pi'}={{\mathcal{P}_T}}(\overline{f})(\mathcal{P}_\pi)\in {{\mathcal{P}_T}}(H_\pi),$$

where $\overline{f}\colon H_\pi\to H_{\pi'}$ is the isomorphism between the free parts of the abelianisations of $G_\pi$ and $G_{\pi'}$ induced by $f$. The conjecture addresses whether the marked polytopes associated to different $(2,1)$-presentations of isomorphic groups are related independently of the choice of presentation; its resolution status is not specified in the source.