Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}_2 (X), W_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. It is known that discrete measures are dense inside $(\mathcal{P}_2 (X), W_2)$ - namely, given any $\mu \in \mathcal{P}_2 (X)$, and $\delta>0$, one can find a discrete measure $\mu_\delta$ with $W_2 (\mu, \mu_\delta)<\delta$.

Now, let $\mu_0, \mu_1 \in \mathcal{P}_2 (X)$, and let $\mu_t$ be a $W_2$ geodesic connecting $\mu_0$ and $\mu_1$ (a.k.a. $\mu_t$ is a [not necessarily unique] displacement interpolation between $\mu_0$ and $\mu_1$). Is the displacement interpolation stable under discrete approximation? That is, can one pick discrete $\mu_{0,n}, \mu_{1,n}$ such that $\mu_{t,n}$ is close to $\mu_t$ for all $t\in[0,1]$?