An improvement of a convergence rate estimate

Let $\xi_1,\xi_2,\dots$ be independent random variables equally distributed with a continuous distribution function$F(x)$. Put
$$ W_n^2=n\int_{-\infty}^\infty[F_n(x)-F(x)]^2\,dF(x), $$
where
$$ F_n(x)=\frac1n\sum_{j=1}^n\delta(x-\xi_j),\quad\delta(x)= \begin{cases} 1,&x>0, \\ 0,&x\le0. \end{cases} $$
Denote by $S(x)$ the distribution function with the characteristic function
$$ s(t)=\prod_{j=1}^\infty(1-2it(\pi j)^{-2})^{-1/2}. $$
In [3], it has been shown that
$$ \Delta_n=\sup_{x\in R^1}|\mathbf P(W_n^2<x)-S(x)|\underset{n\to\infty}\longrightarrow0 $$
not slowlier than $n^{-1/10}$. In the present paper, we obtain a stronger result: for any $\varepsilon>0$ there exists a $c(\varepsilon)$ such that
$$ \Delta_n\le c(\varepsilon)n^{-1/6+\varepsilon},\quad n=1,2,\dots. $$