The higher-power modulus spacing conjecture

For $k\geq 1$ and $Q\in\mathbb N$, define

$$S_{k,Q}=\left\{\frac{a}{q^k}\in\mathbb Q:\gcd(a,q)=1,\ 1\leq a<q^k,\ Q<q\leq 2Q\right\}.$$

The distance from a real number to the nearest integer is denoted by $\|\cdot\|$.

Higher-power modulus spacing conjecture. The local multiplicity of $S_{k,Q}$ at scale $Q^{-k-1}$ should satisfy

$$\max_{x\in S_{k,Q}}\#\left\{x'\in S_{k,Q}:\|x-x'\|<Q^{-k-1}\right\}\ll Q^{\epsilon},$$

where the implied constant depends only on $k$ and $\epsilon$.

This is proposed as an analogue of the square-modulus spacing conjecture for higher powers, and the source explicitly expresses it with less confidence. No resolution is stated.