Duchin–Fisher compact-set conjecture for Thurston-boundary geodesics

Let $S$ be a surface, and let $\alpha$ and $\beta$ be simple closed curves that fill $S$. Let $\mathbf{G}$ be a Teichmüller geodesic ending at $\alpha$ and $\beta$. For sequences $X_n,Y_n\in\mathcal{T}(S)$ converging to $\alpha$ and $\beta$, respectively, in the Thurston compactification, consider the geodesic segments joining $X_n$ to $Y_n$. Duchin–Fisher conjecture. There exists a compact set $\mathcal{K}\subset\mathcal{T}(S)$ such that these geodesic segments intersect $\mathcal{K}$ for all sufficiently large $n$. This is a stickiness statement for Teichmüller geodesics approaching filling boundary points; the paper discusses it as a conjecture of Duchin and Fisher, while proving an analogous convergence result for Gardiner–Masur boundary points.