Suppose $\mu, \nu$ are probability measures on $\mathbb{R}^p$. Let $X,Y$ be i.i.d. samples of size $m$ drawn from $\mu,\nu$. Does there exist a constant $C > 0$ that depends only on the numbers $p$ and $m$, such that the following holds? $$\|\Sigma_X-\Sigma_Y\|_2 < C \cdot W_p(\mu,\nu)$$ Here, $\Sigma_X$ is the covariance matrix of the random variable , $\|A\|_2$ is the 2-norm of a matrix $A$, and $W_p (\mu, \nu)$ is the $p$-Wasserstein distance between $\mu$ and $\nu$.
If there is a weaker result, or a similar result with the Wasserstein distance replaced by something else, I'd love to know them as well.
Update: I proved this own my own, assuming that $X,Y$ are compactly supported. But the proof is a little long. I would be glad to know about an existing reference to this.
See Prop 3.3 in page 14 of this paper. We have: $$\|\Sigma_X-\Sigma_Y\|_2 < 8r \cdot W_p(\mu,\nu)$$ where the each measure is supported on some radius-$r$ ball.