Mahler's rational-counting conjecture for the standard Cantor set

Let $C$ be the standard Cantor set, and let

$$N(s,t)=\left\|\left\{\frac{p}{q}\in C:\gcd(p,q)=1,\ s\leq q\leq t\right\}\right\|.$$

Here $\gcd(p,q)=1$ means that the rational is written in reduced form. Mahler's rational-counting conjecture. For every $\epsilon>0$,

$$N(3^n,3^{n+1})\in O\left(2^{(1+\epsilon)n}\right).$$

This conjecture gives the rational-counting estimate needed to establish the expected intrinsic approximation behavior on $C$ and would imply that the set of intrinsically very well approximable points has zero Hausdorff measure. The source reports computer evidence for it; its stronger possible bound $N(3^n,3^{n+1})\in O(2^n)$ is stated not to hold in forthcoming work, while the displayed conjecture itself is presented without a resolution.