Perelman's classification conjecture for compact ancient non-collapsed Ricci flows on $S^3$

Let $(S^3,g(t))$ be a compact ancient $\kappa$-noncollapsed solution to the three-dimensional Ricci flow

$$\frac{\partial}{\partial t}g_{ij}=-2R_{ij},$$

existing for $t\in(-\infty,T)$ and shrinking to a round point at $T$. Perelman's conjecture. The solution $g(t)$ is either a family of contracting spheres or Perelman's rotationally symmetric ancient solution on $S^3$. Perelman's solution is a type-II ancient solution that is non-collapsed and has the Bryant soliton and the round cylinder $S^2\times\mathbb{R}$ as backward limits; the conjecture would classify compact ancient $\kappa$-noncollapsed three-dimensional Ricci flows relevant to singularity analysis.