Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^d$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.

My question is the following: Let $X$ be a random variable defined on some probability space (rich enough) with law $\mu$, could we find a measurable function $f:\mathbb R^d\times \mathbb R^d\to\mathbb R^d$ and a random variable $G$ independent of $X$ s.t.

$$Y:=f(X,G)~\sim~\nu~~~~~~ \mbox{ and }~~~~~~ \mathbb E[|X-Y|]~\le ~2d~?$$

Some thoughts: Let $d_0:=\rho(\mu,\nu)$, where $\rho(\cdot,\cdot)$ denotes the Prokhorov distance. Then we have a measurable function $f_0:\mathbb R^d\times \mathbb R^d\to\mathbb R^d$ and a random variable $G_0$ independent of $X$ s.t.

$$Y_0:=f_0(X,G)~\sim~\nu~~~~~~ \mbox{ and }~~~~~~ \mathbb P[\{|X-Y|\ge2d_0\}]~\le ~2d_0.$$

The above construction is from the paper On a representation of random variables by Skorokhod, but I can't find this paper.

Any answer, help or comment is highly appreciated. Thanks a lot!