My question comes form a potion of the long review paper, which is attached below

In the set-up, $sigma_1$ and $sigma_2$ are possibly different, constant diffusion matrices. To my knowledge, if we take the (“cheap”) synchronous coupling $B^1_t = B^2_t$, then we can estimate $mathbb{E}(|X^1_t – X^2_t|^2)$ using for instance Ito isometry or BDG inequality. However, what will be the error estimate $mathbb{E}(|X^1_t – X^2_t|^2)$ under this new (“the best”) coupling? The reference paper FH16 published on *The Annals of Probability* is overwhelming long and it is also too technical for me. Thanks for any help!