Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-Wasserstein space $\mathcal{W}(X)$ as a convex subset of $\mathcal{F}(X)$ via the usual embedding: $$ \mathcal{W}(X)\ni \mu \mapsto \left[f\mapsto \int_{u\in X} f(u) [\mu-\delta_{x}](du)\right] \in \operatorname{Lip}_0(X)' =:\mathcal{F}(X); \qquad \boldsymbol{(1)} $$ where the functions $f$ on the right-hand side belong to $\operatorname{Lip}_0(X)$ (so $f(x)=0$, but for generality I have defined them on all of $\operatorname{Lip}(X)$).

THE ASSUMPTION:

We assume that $(X,d,x)$ is such that $\mathcal{F}(X)$ has the bounded approximation property (BAP) (extensive research has been conducted over the last decade on this question, and many "interesting/typical" pointed metric spaces satisfy this assumption). Furthermore, in a broad range of "tame cases" explicit estimates on the operator-norm of the sequence of operators realizing the bounded-approximation property are known (for instance for any closed-subset of an $N$-dimensional Euclidean space the rate is $C\sqrt{N}$ for some universal constant $C$.

THE QUESTION(s): For every $\epsilon>0$, does there exist some Lipschitz $$ \operatorname{P}_{\epsilon}: \mathcal{F}(X)\ni x \mapsto \mathcal{W}(X) $$ satisfying: $$ d_{\mathcal{F}(X)}\left(P_{\epsilon}(x),x\right) \leq \inf_{\mu \in \mathcal{W}(X)}\,d_{\mathcal{F}(X)}\left(\mu,x\right) + \epsilon? $$

Further, can we take the Lipschitz constant of the $P_{\epsilon}$ to be uniformly bounded for small $\epsilon$?