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 ([Lemma 2.1 of this paper][1]).

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


  [1]: https://arxiv.org/pdf/1002.3448.pdf
  [2]: https://mathoverflow.net/q/309372/78539