For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \mathcal{H}(\rho)=\int_{\mathbb{R}^d}\rho\log \rho \,dx $$ be the Boltzmann entropy. It is well-known that $\mathcal{H}$ is lower semi-continuous (l.s.c.) in the Wasserstein metric $W_2$.

Question: Fix $C>0$. Is the sublevel set $\{\rho \in \mathcal{P}^2_{ac}(\mathbb{R}^d) : \mathcal{H}(\rho) \le C\}$ compact in $W_2$?

Thank you so much for your elaboration! Any reference is greatly approciated!