Existence of embedded immortal solutions for higher-order area-preserving curvature flows

Let $m\geq1$. Consider the area-preserving curvature flow of order $m$ for embedded initial curves with length $L_0$ and enclosed area $1$. Immortal-solution conjecture. For every $L_0\geq\sqrt{4\pi}$, there exists an embedded initial curve $\gamma_0$ of length $L_0$ and enclosed area $1$ such that the solution starting from $\gamma_0$ is immortal. This extends the corresponding property known for the $m=0$ area-preserving curve-shortening flow; for $m\geq1$, convexity is not preserved, so a substantially new method beyond the nearly circular regime is required.