Estimates for generalized Dudley's metrics in spaces of finite-dimensional distributions

For arbitrary random vectors $X$ and $Y$ with values in $R^k$ and for any integer $m\ge 1$, the following inequality is proved:
$$ \pi^{m+1}(X,Y)\le c\omega_{m-1}(X,Y). $$
Here $\pi$ is the well-known Lévy–Prohorov metric, $\omega_{m-1}$ is a multidimensional analogue of metrics studied by N. Grigorevski\v i and I. Šiganov [1] and $c$ is a constant depending on $m$ and $k$.