Giga's conjecture on global existence of embedded curve diffusion flows

Let $\gamma:\mathbb{S}\times[0,T)\rightarrow\mathbb{R}^2$ be a curve diffusion flow with smooth initial data $\gamma_0$. Assume that

$$\gamma(\cdot,t)\text{ is an embedding for each }t\in[0,T).$$

Giga's conjecture. Then $T=\infty$.

If true, this would imply that every embedded curve diffusion flow converges smoothly and exponentially fast to a round circle. The conjecture is presented as open; it concerns whether embedded curve diffusion flows can develop finite-time singularities.