Finite generation for the K(n)-local sphere

We prove that the homotopy groups of the $K(n)$-local sphere are finitely generated ℤ${}_p$-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after $K(n)$-localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.