The squaring conjecture for subGaussian optimal transport map estimation

Let $\mu$ and $\nu$ be the source and target measures in the setting of the paper's subGaussian convergence theorem. Let $\hat{T}_n$ be the empirical entropy-regularized optimal transport map based on $n$ samples, and let $T_\varepsilon$ be the population entropy-regularized map. The norm is the $L^2(\mu)$ norm and expectations are over the sampling procedure.

Squaring conjecture. In this setting,

$$\mathbb{E}\left[\left\lVert\hat{T}_n-T_\varepsilon\right\rVert_{L^2(\mu)}^2\right]\lesssim n^{-1/3},$$

where the implicit constant may depend on every parameter other than $n$.

The conjecture would provide a squared-error analogue of the subGaussian convergence result, overcoming a technical obstacle involving conditional application of a Wasserstein--relative-entropy inequality. The source says that it may require a different analysis and that, if the variance conjecture holds, the rate should improve to $n^{-1/2}$; the stated result itself remains open.