Random-point conjecture for sums of square roots modulo one

For integers $r\ge 1$ and real numbers $n\ge 1$, $0<\eta\le 1/2$, define

$$A_{2r}(n,\eta):=\#\left\{(a_1,\ldots,a_r,b_1,\ldots,b_r):a_i,b_j\sim n,\ \left\lVert\sum_{i=1}^r\sqrt{a_i}-\sum_{j=1}^r\sqrt{b_j}\right\rVert\le\eta\right\}.$$

Here $a_i,b_j\sim n$ denotes the same dyadic range as in the source, and $\lVert\cdot\rVert$ is distance to the nearest integer. Random-point conjecture. For every fixed integer $r\ge 1$ and every fixed $\varepsilon>0$, uniformly for $0<\eta\le 1/2$, one has

$$A_{2r}(n,\eta)\ll_{r,\varepsilon} n^{r+\varepsilon}+n^{2r+\varepsilon}\eta.$$

The conjecture is motivated by treating the fractional parts of $\sum_{i=1}^r\sqrt{a_i}$ as $n^r$ random points in $[0,1)$. It would provide the higher-moment counting input needed, via the displayed reduction in the source, to estimate the even moments $M_{2r}(H,n)$; no resolution is given here.