Distribution of an analog of Sherman's statistics under rank-censored observations

Let $U_n(1),\dots,U_n(n)$ be a variational series constructed from a sequence of $n$ aggregate-independent random variables distributed uniformly on $(0,1)$. Let $k_0=0$, $k_1,\dots,k_m,k_{m+1}=n+1$ be an increasing sequence of nonnegative integers, $\lambda_r=k_{r+1}-k_r$, $r=0,\dots,m$ and
$$ \xi_n=\frac12\sum^m_{r=0}\Bigr|U_n(k_{r+1})-U_n(k_r)-\frac{k_{r+1}-k_r}{n+1}\Bigl|. $$
Under certain restrictions on the numbers $\lambda_r=k_{r+1}-k_r$, in this paper we have shown the asymptotic normality (with an appropriate norming) of the quantity $\xi_n$ as $n,m\to\infty$ such that $\lim\sup(m/\sqrt n)\to\infty$.