Generalization of Waring's problem for nine almost proportional cubes

An asymptotic formula is obtained for the number of representations of a sufficiently large natural $N$ as a sum of nine cubes of natural numbers $x_i$, $i=\overline{1,9}$, satisfying the conditions
$$|x_i^3-\mu_iN|\le H, \mu_1+\ldots+\mu_9=1 H\ge N^{1-\frac1{30}+\varepsilon} ,$$
where $\mu_1,\ldots,\mu_9$ — positive fixed numbers. This result is a strengthening of E.M.Wright's theorem.