Cyclic pursuit convergence conjecture on compact Riemannian manifolds

Let $M$ be a compact Riemannian manifold. For cyclic pursuit curves $b_i(t)$, let $\beta^t$ be the piecewise geodesic loop obtained by concatenating the shortest geodesics between consecutive agents. Cyclic pursuit convergence conjecture. The family $\beta^t$ either collapses to a point in finite time or converges to a closed geodesic as $t\rightarrow\infty$. This extends the finite-time collision behavior of Euclidean cyclic pursuit to compact manifolds while allowing nontrivial homotopy classes; it is proved in the paper for nonpositively curved compact manifolds, but remains open in general.