Conjecture on long-period rational points in the Cantor set

Let $C$ be the Cantor set and let $P(p/q)$ denote the period of the ternary expansion of the rational number $p/q$. For

$$S_n:= \{p/q\in C:\gcd(p,q)=1,\;3^{n-1}\leq q<3^n\},$$

set

$$S_n^{(K)}:= \{p/q\in C:\gcd(p,q)=1,\;3^{n-1}\leq q<3^n,\;P(p/q)\leq K\log(q)\}.$$

Long-period conjecture. For every $K>2/\log(3/2)$, one has $S_n^{(K)}=S_n$ for all sufficiently large $n$; in particular,

$$\#\bigl(S_n\setminus S_n^{(K)}\bigr)=o(1).$$

This is presented as a stronger conjecture supporting the preceding reduction. If true, it implies the weaker assertion that there exists some finite $K$ with the complement bounded by $O\bigl(2^{n(1+\varepsilon_1)}\bigr)$, and hence would imply the cited zero-measure consequence for very well intrinsically approximable points.