Conjecture on dynamically stable solutions for curve diffusion, elastic, and ideal flows

A planar curve $\gamma$ is dynamically stable if there exists an $\epsilon>0$ such that every curve $\eta$ with $||\gamma-\eta||_{C^\infty}<\epsilon$ generates a solution trajectory converging, modulo similarity transformation, to $\gamma$. The curve diffusion flow, elastic flow, and ideal flow are geometric flows for planar curves.

Stability classification conjecture. The only dynamically stable solutions are circles for the curve diffusion flow; multiply-covered lemniscates of Bernoulli and multiply-covered circles for the elastic flow; and multiply-covered ideal lemniscates and multiply-covered circles for the ideal flow.

The conjecture concerns the dynamical stability of homothetic solutions. Its ideal-flow assertion presupposes the rigorous discovery of an ideal lemniscate, a self-similarly expanding figure-eight-type solution, which the source states has not yet been established.