Zippers-to-flows conjecture for hyperbolic groups

Let $G$ be a hyperbolic group with boundary $\partial_\infty G$ homeomorphic to $S^2$, and suppose that $Z^\pm\subset\partial_\infty G$ is a minimal $G$-zipper. Let $P\subset Z^+\times Z^-$ be the closure of the set of pairs $p^+\times p^-$ such that $p^\pm$ are the fixed points of an infinite-order element $g\in G$ fixing exactly one point in each of $Z^+$ and $Z^-$. Zippers-to-flows conjecture. The space $P$ is homeomorphic to $\mathbb R^2$. This conjecture seeks a planar model for the paired fixed-point data associated with a minimal zipper; the source presents it as an extension of the zipper constructions from closed hyperbolic 3-manifold groups to general hyperbolic groups.