Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality

$$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$

where $n_i$ are positive integers with $M<n_i\leq 2M$ for all $1\leq i\leq 4$. Let us denote this quantity by $N(M,\delta)$.

There is a paper by O.Robert and P.Sargos which provides the bound for a bit more general quantity. In our case their work gives

$$N(M,\delta)\ll M^{2+o(1)}+\delta M^{4+o(1)}.$$

I think, some heuristics suggest that when $\delta\gg 1$ this bound is more or less optimal. But I'm interested in the case when $\delta=o\left(\frac{1}{M}\right)$. So, my question is: can the bound of Robert and Sargos be improved for small $\delta$ and when should we expect that $N(M,\delta)\ll M^{2+o(1)}$? For example, we always have $N(M,M^{-2})\ll M^{2+o(1)}$ but can we do better (i.e. prove this bound for larger $\delta$) or at least are there any conjectures about this?