On the convergence speed of distribution of maximum sums of independent random variables

Let $\xi_n$, $n=1,2,\dots,$ be a sequence of independent identically distributed random variables with $\mathbf M\xi=0$. Put $\sigma^2=\mathbf D\xi_1$, $c_3=\mathbf M|\xi_1|^3$, $S_n=\sum_{i=1}^n\xi_i$, $S_n^-=\max\limits_{1\le i\le n}S_i$, $\overline F_n(x)=\mathbf P(\overline S_n<x)$.
The following estimate is obtained: there exists an absolute constant $K$ such that
$$ \sup_{0\le x<\infty}|\overline F_n(x\sigma\sqrt n)-\biggl(\frac2\pi\biggr)^{1/2}\int_0^xe^{-u^{2/3}}\,du|<K\frac{c_3^2}{\sigma^6\sqrt n}. $$