Background. The Skorohod representation theorem says that if $X_n \to_{\mathrm{d}} X$ (i.e. $X_n$ converges to $X$ in distribution as $n \to \infty$), then we can construct a new probability space with a sequence $Y_n$ and another variable $Y$ such that $X_n$ and $Y_n$ have the same law and $Y_n \to_{\mathrm{a.s.}} Y$ (i.e. convergence almost surely).
Question. Suppose that instead of convergence in distribution, we have convergence of distributions in Wasserstein $p$-distance $W_p$ (see below), which we'll denote by $X_n \to_{W_p} X$. Can we construct a new probability space with a sequence $Y_n$ and another variable $Y$ such that $X_n$ and $Y_n$ have the same law and $Y_n \to_{L^p} Y$? (And perhaps $Y_n \to_{\mathrm{a.s.}} Y$ as well?) If so, does this result have a standard name/reference?
(If it makes things easier, let's assume we're just working with $\mathbb{R}$-valued random variables and measures on $\mathbb{R}$.)
Definitions. The Wasserstein $p$-distance between two measures $\mu$ and $\nu$, denoted $W_p(\mu, \nu)$, is the minimum possible value of $\Vert X - Y\Vert_p = \mathbf{E}[|Z_\mu - Z_\nu|^p]^{1/p}$, where $Z_\mu \sim \mu$ and $Z_\nu \sim \nu$ are defined on a common probability space. (The minimizing pair $Z_\mu, Z_\nu$ is called an "optimal coupling" of $\mu$ and $\nu$.) This $W_p$ is a metric that induces a topology. By "convergence of distributions in $W_p$", I mean convergence of the law of $X_n$ to the law of $X$ in the topology induced by $W_p$.
Intuition. I think it should suffice if we had optimal couplings $Z, Z_n$ between the laws of $X$ and $X_n$, all on outcome space $\mathbb{R} \times \mathbb{R}$ with possibly different probability measures, such that $$\begin{aligned} Z(\psi, \omega) &= \psi, \\ Z_n(\psi, \omega) &= \psi + \omega. \end{aligned}$$ Given these, we could stitch them together into a probability measure on $\mathbb{R} \times \mathbb{R}^{\mathbb{N}}$, i.e. the space of sequences $(\psi, (\omega_0, \omega_1, \dots))$, such that $$\begin{aligned} Y(\psi, \omega) &= \psi, \\ Y_n(\psi, \omega) &= \psi + \omega_n, \end{aligned}$$ and the law of $(Y_n, Y)$ is the same as the law of $(Z_n, Z)$.