A subquadratic rational-counting conjecture for the standard Cantor set

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

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

Subquadratic rational-counting conjecture. There exists a constant $\gamma<2$ such that

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

This is introduced as a weaker variant of the preceding rational-counting conjecture, motivated by the extrinsic approximation problem for rationals in the Cantor set. The source gives no resolution of this weaker assertion.