On Vu's restricted box estimate in Waring's problem

In 2000, Vu proved that the number of solutions of $x_1^k+\cdots +x_s^k=N$ in an arbitrary box satisfies the expected Hardy–Littlewood upper bound with a power-saving error term, for $s\ge O(8^k{k}^3)$. We show that one may take $s\ge k^2-k+O(\sqrt{k})$.