The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$$
where $\Gamma(\nu_{1},\nu_{2})$ is the set of all couplings between $\nu_1$ and $\nu_2$. For $X=\mathbb{R}^d$ and $d$ being the Euclidean distance the optimal transport between $\nu_{1}=N(m,V)$ and $\nu_{2}=N(n,U)$ is well known for $p=2$, see Givens and Shortt. On the other hand $W_1$ has stronger Kantorovich duality. So my question is: is there a chance that both things are basically the same and we can have the best of both worlds?
More precisely: does there exist a constant $C > 0$ such that for any dimension $d$, $m = 0, n = 0$ and any $U, V$ we have
$$\frac{W_2(\nu_{1},\nu_{2})}{W_1(\nu_{1},\nu_{2})} \le C?$$
I assume that there is a counterexample but I checked the dimension $1$ and here it's fine.
Also this question is a mod of an old one.
