Let $\mathcal{M}_1(\mathbb{R}^{n})$ be the space of probability measures on $\mathbb{R}^{n}$.
Given two probability distributions $\mu,\nu$ on $\mathbb{R}^{2n}$, let their marginals be denoted $\mu_i,\nu_i \in \mathcal{M}_1(\mathbb{R}^{n})$, $i=1,2$. The 2-Wasserstein distance (associated to Euclidean metric) between the 1-st marginals is
$$W^2_2(\mu_1,\nu_1)=\inf_{\gamma\in \Pi(\mu_1,\nu_1)}\int \|q-\tilde{q}\|^2 d\gamma(q,\tilde{q})~~~~~~~~~~(1)$$
where $\Pi(\mu_1,\nu_1)$ is the set of joint distributions between $\mu_1,\nu_1$. My question is : $$\textbf{Can this be written as an O.T problem between the full distributions?}$$
I.e defining $c:\mathbb{R}^{2n}\times \mathbb{R}^{2n}\to \mathbb{R}$ as $$c\big((q,p),(\tilde{q},\tilde{p})\big)=\begin{cases} \infty, &\text{if} ~p\neq \tilde{p} \\ \|q-\tilde{q}\|^2,& \text{otherwise.} \end{cases}$$
Does $(1)$ coincide with
$$\inf_{\pi\in \Pi(\mu,\nu)} \int c\big((q,p),(\tilde{q},\tilde{p})\big) d\pi(q,p,\tilde{q},\tilde{p})$$
if the marginals are equal $\mu_1=\nu_1$.