On optimal matching of Gaussian samples

Let $X_1,\dots,X_n$ be independent random variables with common distribution the standard Gaussian measure $\mu$ on $\mathbb R^2$, and let $\mu_n=\frac1n\sum_{i=1}^n\delta_{X_i}$ be the associated empirical measure. We show that, for some numerical constant $C>0$,
$$ \frac1C\frac{\log n}n\leq\mathbb E(\mathrm W_2^2(\mu_n,\mu))\leq C\frac{(\log n)^2}n $$
where $\mathrm W_2$ is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.