The Gaussian $n^{-1}$ convergence conjecture for entropy-regularized optimal transport maps

Let $\mu$ and $\nu$ be mean-zero multivariate Gaussian measures. Let $\hat{T}_n$ be the empirical entropy-regularized optimal transport map based on $n$ samples, and let $T_\varepsilon$ be the corresponding population map. The norm is the $L^2(\mu)$ norm and expectations are over the sampling procedure.

Gaussian $n^{-1}$ convergence conjecture.

$$\mathbb{E}\left[\left\lVert\hat{T}_n-T_\varepsilon\right\rVert_{L^2(\mu)}^2\right]\lesssim \frac{1}{n}.$$

For compactly supported measures, an $n^{-1}$ convergence rate is known, while the paper reports empirical evidence for the same rate for multivariate Gaussian source and target measures. Establishing this rate explicitly for mean-zero multivariate Gaussians remains open.