Let $S$ be a metric space and denote the set of probability measures on $S$ by $\mathcal{P}(S)$. Fix $\mu\in \mathcal{P}(S)$ and denote the law of $N\geq 1$ i.i.d samples $X=(X_1,\ldots,X_N)$ from $\mu$ as $\mu_N$. Do we have the following for $p\geq 1$: \begin{align} \inf_{\nu \in \mathcal{P}(S)} \mathbb{E}_{X\sim \mu_N}[\mathcal{W}_p(\frac{1}{N}\sum_{i=1}^N \delta_{X_i}, \nu)] = \mathbb{E}_{X\sim \mu_N}[\mathcal{W}_p(\frac{1}{N}\sum_{i=1}^N \delta_{X_i}, \mu)] \,? \end{align} In plain words: Is the empirical distribution, on average, closest to its underlying distribution?

So far, I did neither succeed in proving this nor in finding some reference, even though there are many articles studying the asymptotics of the r.h.s.. Does anyone know whether this is true, at least in some simplified scenario ($S=[0,1], p=1$ e.g.)?