In this paper https://arxiv.org/abs/2102.00356, I try to prove the Theorem 2.3 on page 4. It seems that there is no proof on the paper.

They introduce a new Wasserstein correlation for $\pi\in \Pi(\mu,\nu)$ the set of coupling of two probability measures $\mu$ and $\nu$ on a Polish space $(X,d)$ $$ W(\pi)=\frac{\int W_1(\pi_{x_1},\nu)\mu(dx_1)}{\int d(y,z)\nu(dy)\nu(dz)}, $$ where $\pi$ has a $\mu$-a.s. unique disintegration w.r.t. the first coordinate and $W_1$ is the 1-Wasserstein distance.

Let $\hat{\pi}^N$ be an AW-consistent estimator of $\pi$ (i.e., $\hat{\pi}^N$ is a strongly consistent estimator of $\pi$ under adapted (or called nested) Wasserstein distance $AW(\pi, \hat{\pi}^N)\to 0 $ a.s.).

How to prove that $W(\hat{\pi}^N)$ is a consistent estimator of $W(\pi)$?