Bellissard's Gap Labeling Conjecture for Cantor actions

Let $\Sigma$ be a Cantor set, meaning a compact, totally disconnected metric space without isolated points, and let $\mathbb{Z}^{n}$ act freely and minimally on $\Sigma$, with invariant probability measure $\mu$. The measure induces a trace $\mu:C(\Sigma)\to\mathbb{C}$, while the action induces the crossed-product algebra $C(\Sigma)\rtimes\mathbb{Z}^{n}$ and its trace $\tau_{\mu}:C(\Sigma)\rtimes\mathbb{Z}^{n}\to\mathbb{C}$. Write the induced maps on K-theory with the same symbols. Bellissard's Gap Labeling Conjecture. One has

$$\mu(K_{0}(C(\Sigma)))=\tau_{\mu}(K_{0}(C(\Sigma)\rtimes\mathbb{Z}^{n})).$$

The conjecture identifies the possible spectral gaps of associated Schrödinger operators with the range of the canonical trace on the K-theory of the crossed-product algebra. The paper presents a proof using an appropriate version of Connes' index theorem for foliated spaces, so the claim is solved in this source.