The spacing conjecture for reduced fractions with square moduli

Let $S_Q$ be the set of reduced fractions $\frac{a}{q^2}$ with $Q<q\leq 2Q$ and $1\leq a<q^2$. For real numbers, write $\|x\|$ for the distance from $x$ to the nearest integer. The maximum number of elements of $S_Q$ within distance $Q^{-3}$ of any element should satisfy

Spacing conjecture for square moduli.

$$\max_{x\in S_Q}\#\left\{x'\in S_Q:\|x-x'\|<Q^{-3}\right\}\ll Q^{\epsilon},$$

where the implied constant depends on $\epsilon$ alone.

This conjecture asserts that reduced fractions with square denominators have essentially optimal local spacing. It is motivated by the expected $Q^{-3}$ average spacing among the approximately $Q^3$ elements of $S_Q$; the source gives numerical evidence but no resolution.