For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the Wasserstein metric $W_p$. Now we fix $p \in (1, \infty)$.

Are there Lipschitz maps $I_p^n : \mathcal P_1 (\mathbb{R^d}) \to \mathcal P_p (\mathbb{R^d})$ for $n \in \mathbb N$ such that $\lim_{n \to \infty} W_1(\mu, I^n_p (\mu))=0$ for all $\mu \in \mathcal P_1 (\mathbb{R^d})$?

Thank you so much for your elaboration!