Penalty Pairs with Regular Penalised Maxima

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. Intrinsic test functions on $\mathcal{D}$ are those of that definition, and local maxima relative to $\mathcal{D}$ are taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i). For $\chi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and positive $\lambda\in\mathbb{R}$, $\chi-\lambda\mathcal{E}$ is the function on $\mathcal{D}$ with value $\chi(\mu)-\lambda\,\mathcal{E}(\mu)$ at $\mu$.

(Regular penalised maxima)(i) The penalty pair has regular penalised maxima if for every intrinsic test function $\chi$ on $\mathcal{D}$ and every positive $\lambda\in\mathbb{R}$, every $\mu\in\mathcal{D}$ at which $\chi-\lambda\mathcal{E}$ has a local maximum relative to $\mathcal{D}$ belongs to $\mathcal{D}_{\Sigma}$.