Let $n\in\mathbb N$, let $\lambda>0$, and let $\nu=e^{-V}\,d\mathcal L^n$ be a Borel probability measure on $\mathbb R^n$, where $V\in C^2(\mathbb R^n)$ satisfies
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\begin{align*}
J(\nabla V)_x\ge \lambda I
\end{align*}
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as quadratic forms for every $x\in\mathbb R^n$. Let $\mu=f\nu$ be a probability measure such that $\mu$ satisfies the smoothness, positivity, finite-second-moment, finite-entropy, finite-Fisher-information, and decay hypotheses in the Otto-Villani HWI inequality. Define