$\kappa$-solutions with the round cylinder as an asymptotic shrinker

We show that $\kappa$-solutions to the Ricci flow in dimensions $n\ge 4$ whose asymptotic shrinking Ricci soliton is the round cylinder ${\mathbb{S}}^{n-1}\times \mathbb{R}$ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman’s ancient solution.