The variance conjecture for entropy-regularized optimal transport maps

Let $\mu$ and $\nu$ be subGaussian measures. Let $T_{1,m}$ denote a single map term based on $m$ samples, and let $\hat{T}_n$ denote the resulting estimator based on $n=km$ samples, where $k$ is the number of terms. The norm is the $L^2(\mu)$ norm and expectations are over the sampling procedure.

Variance conjecture.

$$\mathbb{E}\left\lVert T_{1,m}-\mathbb{E}[T_{1,m}]\right\rVert_{L^2(\mu)}^2\lesssim \frac{1}{m}.$$

Equivalently,

$$\mathbb{E}\left\lVert\hat{T}_n-\mathbb{E}[\hat{T}_n]\right\rVert_{L^2(\mu)}^2\lesssim \frac{1}{k}\cdot\frac{1}{m}=\frac{1}{n},$$

with a constant independent of $m$, $k$, and $n$.

The conjecture concerns the empirically observed variance decay for subGaussian source and target measures. A positive answer would improve the variance contribution in the estimator's bias-variance analysis from order $k^{-1}$ to order $n^{-1}$, and would improve the resulting convergence rates. It remains open according to the source.