Log-normal approximation for the Wasserstein distance between Gaussian samples

Let $\mathbf{W}^{(n)}=\mathrm{W}_2^2(\hat\mu^{1,(n)},\hat\nu^{1,(n)})$ be the squared Wasserstein distance between two empirical samples of size $n$ from the standard normal distribution, and let $\mathcal{LN}(\mu_n,\tau_n)$ denote a log-normal distribution with parameters $\mu_n$ and $\tau_n$. Log-normal approximation conjecture. For $n$ large enough, the distribution of $\mathbf{W}^{(n)}$ can be approximated by $\mathcal{LN}(\mu_n,\tau_n)$. This approximation is proposed to obtain an explicit usable description of the statistic for sufficiently large samples, typically $n\ge 20$; the source provides no resolution of the conjecture.