The rate of convergence of the Smirnov–Mises statistic's distribution

We consider $n$ independent random variables with a continuous distribution function $F(x)$ and empirical distribution function $F_n(x)$. Put
$$ \omega_n^2=n\int_{-\infty}^\infty(F_n(x)-F(x))^2\,dF(x) $$
and
\begin{gather*} S(z)=\lim_{n\to\infty}\mathbf P\{\omega_n^2<z\}, \\ \Delta_n=\sup_{-\infty<z<\infty}|\mathbf P\{\omega^2<z\}-S(z)|. \end{gather*}
Many papers dealt with the estimate: For each $\varepsilon>0$, there exists a $b(\varepsilon)$ such that
$$ \Delta_n<b(\varepsilon)n^{-a+\varepsilon} $$
for $n=1,2,\dots$.
The inequality (1) is proved for $a=1/10$ [7], $a=1/6$ [8], $a=1/4$ [9], $a=1/3$ [10].
In the present paper, we obtain (1) for $a=1/2$.