Global asymptotic-shape conjecture for the free boundary free elastic flow

Let $\gamma:[-1,1]\times[0,\infty)\to\mathbb{R}^2$ be a free boundary free elastic flow. Let $\overline{\gamma}(t)$ be its barycentre and $\mathcal{L}[\gamma(\cdot,t)]$ its length, and set

$$\hat\gamma(\cdot,t)=\frac{\gamma(\cdot,t)-\overline{\gamma}(t)}{\mathcal{L}[\gamma(\cdot,t)]}.$$

Global asymptotic-shape conjecture. The rescaled flow $\hat\gamma(\cdot,t)$ converges as $t\to\infty$ exponentially fast in the smooth topology to a limit $\gamma_\infty$. Furthermore, $\gamma_\infty$ is either the horizontal line $[-1,1]\times\{0\}$, a rescaled Euler's rectangular elastica with $m$ half-periods, or a self-similar expander for the limiting free boundary free elastic flow.

The conjecture describes the possible global asymptotic shapes after normalizing by the curve length and recentering at the barycentre. The cited context identifies semicircles, circles and their multiple coverings, as well as half-periods and full periods of the Lemniscate of Bernoulli, as candidate self-similar expanders; no resolution is supplied here.