Stability of Llarull's theorem in all dimensions

Llarull's theorem characterizes the round sphere $S^n$ among all spin manifolds whose scalar curvature is bounded from below by $n(n-1)$. In this paper we show that if the scalar curvature is bounded from below by $n(n-1)-\epsilon$, the underlying manifold is $C^0$-close to a finite number of spheres outside a small bad set. This completely solves Gromov's spherical stability problem and is the first instance of a scalar curvature stability result that both holds in all dimensions and is stated without any additional geometrical or topological assumptions.