Elliott's conjecture on geodesic curvature flow of noncontractible curves on a cone

Let $\mathcal M$ be a cone and let a closed curve on $\mathcal M$ be not homotopic to a point on $\mathcal M$. Elliott's conjecture. Under geodesic curvature flow, the curve will converge to the apex in finite time. This conjecture concerns the finite-time behavior of geodesic curvature flow for noncontractible closed curves on a cone; the source presents it as a conjecture based on numerical experiments, and no resolution is given here.