Sticky-geodesic conjecture for Teichmüller geodesics with curve endpoints

Let $\gamma$ be a Teichmüller geodesic with endpoints $\gamma^-$ and $\gamma^+$ in the Thurston boundary, and call it sticky when it satisfies the sticky-geodesic condition that geodesics joining sequences approaching its endpoints pass through a fixed compact region. Sticky-geodesic conjecture. Every Teichmüller geodesic whose endpoints are simple closed curves is sticky.

The paper proves that sticky geodesics have endpoints separable by stars. If this conjecture holds, then curve-complex distance at least three would imply star distance at least two, yielding a $(2,0)$-quasiisometry between the star and curve-complex metrics. The conjecture remains open in the supplied text.