Conjecture on centroidal critical points under curve-shortening flow

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow. Denote by $n_U(t)$ the number of critical points of its radial function relative to the fixed point $U$, and by $C(t)$ its centroid. Centroidal critical-point conjecture. At the maximal time $t_{\max}$, one has

$$n_U(t_{\max})=4,$$

and the four remaining trajectories meet at right angles at $U=C(t_{\max})$. The conjecture concerns the limiting configuration of critical-point trajectories; numerical evidence is provided in the paper, while the asserted limiting count and orthogonality are not proved there.