In this paper https://arxiv.org/abs/2102.00356, 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.

The author introduce a plugin estimator. Given iid random samples $(X_1,Y_1),\dots, (X_N,Y_N)$. Let's partition the $[0,1]$ into a disjoint union of a finite number of intervals and let $\phi: [0,1]\mapsto [0,1]$ map each intervals to its center. Let
$$
\hat{\pi}=\frac{1}{N}\sum_{i=1}^N \delta_{(\phi(X_i), \phi(Y_i))}
$$
Define $\Phi:=\{(\phi)^{-1}(\{x\}): x\in \phi([0,1])\}$.

(1) How to understand $\hat{\pi}$? 

(2) As on page 23, why do we have for any Borel set $G\subset [0,1]$
$$
W(\hat{\pi})=\frac{\sum_{G\in \phi} \frac{1}{N}|i\in\{1,\dots,N\} s.t. X_i\in G|\cdot W(\hat{\pi}_G, \hat{\pi}_2)}{\frac{1}{N^2}\sum_{n,m=1}^N|\phi(Y_n)-\phi(Y_m)|}
$$
where $\pi_G$ is the conditional probability
$$
\pi_G(\cdot)=\frac{1}{\pi_1(G)}\int_G \pi_{x_1}(\cdot)\pi_1(dx)
$$

(3) What is $\hat{\pi}_G$? I am confused about this empirical measure.