Perelman's uniqueness conjecture for compact non-collapsed ancient solutions on $S^3$

Let $(S^3,g(t))$ be a compact, ancient $\kappa$-noncollapsed solution to the Ricci flow on $S^3$, where $\kappa$-noncollapsed means that there is some $\kappa>0$ such that every metric ball $B_r$ with $r<\rho$ and $|Rm|\leq r^{-2}$ on $B_r$ has volume at least $\kappa r^3$ on every scale. Perelman's conjecture. The solution $g(t)$ is either a family of contracting spheres or Perelman's solution. This conjecture seeks to classify compact, ancient, non-collapsed three-dimensional Ricci flows; the source presents it as an open conjecture, with Perelman's solution and contracting spheres as the expected possibilities.