Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Now, It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution $\nu$ (Lemma 2.1 of this paper).

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?