Ideal moment conjecture for sums of square roots modulo one

Let $r\geq 1$ be fixed. For $h\sim H$, define the moment $M_{2r}(H,n)$ as in the source. Ideal moment conjecture. For every fixed $\delta>0$ and every $\varepsilon>0$, one expects

$$M_{2r}(H,n)\ll_{r,\varepsilon,\delta}n^\varepsilon\begin{cases}n^rH^{1-2r},&1\leq H\leq n^{1/2-\delta},\Hn^r,&H\geq n^{1/2+\delta}. \end{cases}$$

The two ranges reflect the complementary pointwise and dual-expansion heuristics. The intermediate range $n^{1/2-\delta}<H<n^{1/2+\delta}$ is not covered by the stated conjecture, and the source presents these bounds as expected rather than established.