Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct two continuous martingales $X, Y$ (on suitable probability space) such that
$${\rm Law}(X)={\rm Law}(M),\quad {\rm Law}(Y_1)=\nu,\quad \mathbb E\big[\sup_{0\le t\le 1}|X_t-Y_t|^2\big]\le 8\varepsilon^2?$$
PS : We assume $\mu,\nu$ both admit finite second order moment, and $W_2$ denotes the Wasserstein distance of order $2$.
Some thoughts : As mentioned below, the difficult step is to construct a "continuous" martingale by taking conditional expectation. I did some attempts as follows :
Step 1. Take first a discrete martingale, denoted by $M^n:=(M^n_{t_k})_{0\le k\le n}$ such that ${\rm Law}(M)\approx {\rm Law}(M^n)$;
Step 2. Using the result Can every discrete martingale be embedded in a continuous martingale? where we notice that we approximate further $M^n$ by a continuous martingale $N^n$ whose natural filtration can be quite "Brownian";
Step 3. Approximate $N^n_1$ by $Y_1$ (using Skorokhod representation theorem) and then take conditional expectation of $Y_1$ with respect to the filtration generated by $N^n$.
Can we make thesethe above arguments rigorous?
