Mahler's intrinsic approximation conjecture for the Cantor set

Let ${\mathcal{C}}$ be Cantor's middle-thirds set, equipped with its coin-tossing measure, which assigns equal probability to the digits $0$ and $2$ in the ternary expansion. A rational $p/q\in{\mathcal{C}}$ is understood to have coprime numerator and denominator. Mahler's intrinsic approximation conjecture. For almost every $x\in{\mathcal{C}}$ with respect to the coin-tossing measure, for every $\varepsilon>0$, there are only finitely many rationals $p/q\in{\mathcal{C}}$ such that

$$\left|x-\frac{p}{q}\right|<\frac{1}{q^{1+\varepsilon}}.$$

The conjecture is presented as a consequence of the preceding counting conjecture and concerns intrinsic Diophantine approximation by rationals lying in the Cantor set. The source does not provide a resolution.