Visual-topology conjecture for sublinearly Morse boundaries of graph manifolds

Let $G$ be the fundamental group of a non-positively curved graph manifold, let $X$ be its universal cover, and let $T_{\infty}$ be the Bass–Serre tree of the graph-of-groups decomposition. Let $\mathcal{T}(G,X,T_{\infty})$ be the topology induced by the action of $G$ on $T_{\infty}$, and let $\partial_{\kappa}X$ be the $\kappa$-Morse boundary. Graph-manifold visual-topology conjecture. The topology $\mathcal{T}(G,X,T_{\infty})$ coincides with the visual topology on $\partial_{\kappa}X$. The paper describes proposed approaches via curtain and median topologies, but controlling flats crossing infinitely many pieces remains an obstacle; the conjecture is open.