Uniqueness of tangent flows in free boundary flow

We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form ${\mathbb{S}}^{n-k}\times {\mathbb{R}}_{+}^k$ or ${\mathbb{S}}_{+}^{n-k}\times {\mathbb{R}}^k$, then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.