Improved rates of convergence for the multivariate Central Limit Theorem in Wasserstein distance

We provide new bounds for the rate of convergence of the multivariate Central Limit Theorem in Wasserstein distances of order $p\ge 2$. In particular, we obtain what we conjecture to be the asymptotically optimal rate whenever the density of the summands admits a non-zero continuous component and has a non-zero third moment.