The Gaussian Free-Energy Pair is Wasserstein-Closed

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$, a penalty pair by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair.

(Wasserstein-closed)¶ The pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is Wasserstein-closed.