Define :
$\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^d$.
$\Pi(\mu,\nu)$ the set of probability measures on $\mathbb{R}^d \times \mathbb{R}^d$ with finite second moments and with marginals $\mu,\nu$.
Let $ \{ \mu_k \}_{k=1}^\infty, \{ \nu_k \}_{k=1}^\infty \subseteq \mathcal{P}(\mathbb{R}^n)$ such that $ \mu_k,\nu_k$ have finite second moments, and weakly converge to $\mu,\nu \in \mathcal{P}(\mathbb{R}^n)$ respectively, also having finite second moments. Associated let $\{\pi_k\}_{k=1}^\infty \subseteq \Pi(\mu_k,\nu_k)$.
Is it true that upon extraction of a subsequence $\pi_k \to \pi \in \Pi(\mu,\nu)$ weakly?