Conjecture on asymptotic convergence for immortal elastic and ideal flows

Let $\gamma:[0,\infty)\to\mathbb{R}^2$ be an immortal solution of the elastic flow or the ideal flow with generic initial data.

Asymptotic convergence conjecture. The curve $\gamma(\cdot,t)$ converges exponentially fast in the smooth topology to a multiply-covered lemniscate or circle.

This is proposed as an analogue for the elastic and ideal flows of a result cited in the source. It is presented as a future target after the dynamical-stability conjecture and remains open.