Basic homogeneity in the class of zero-dimensional compact spaces

We show that under the continuum hypothesis there is a compact zero-dimensional space which admits a base of pairwise homeomorphic clopen subsets but it is not an h-homogeneous space (i.e. not all of its nonempty clopen subsets are homeomorphic), partially answering a question of M.V. Matveev. Under Jensenʹs ⋄ principle, we can even make the space hereditarily separable and hence, by a result of Matveev, an S-space.