Spacing of unit fractions

Let $r\geqslant 1$ be an integer, and let $X$ be a positive parameter. Spacing of unit fractions. The number of $x_1,\dots,x_{2r}\in[X,2X)$ satisfying

$$\left|\frac{1}{x_1}+\dots+\frac{1}{x_r}-\frac{1}{x_{r+1}}-\dots-\frac{1}{x_{2r}}\right|\leqslant\frac{1}{X^{r+1}}$$

is $\ll_{r,\varepsilon}X^{r+\varepsilon}$ for all $\varepsilon>0$. This is a further conjecture concerning the spacing and additive relations among reciprocals of integers; the source provides no resolution, so it remains open.