The area-invariance conjecture for concentric circular 3-Poncelet pairs

Let $(\mathcal{C},\mathcal{D})$ be a 3-Poncelet pair consisting of a circle $\mathcal{C}$ and a central conic $\mathcal{D}$, and let $\mathcal{P}$ be a family of Poncelet triangles associated with this pair. For a triangle $\triangle ABC\in\mathcal{P}$, write $\mathrm{Area}(\triangle ABC)$ for its area.

Area-invariance conjecture. The area $\mathrm{Area}(\triangle ABC)$ is independent of the choice of $\triangle ABC\in\mathcal{P}$ if and only if $\mathcal{C}$ and $\mathcal{D}$ are concentric circles.

This conjecture is motivated by numerical evidence and extends the preceding result for pairs in which the center of the circle is at one of the foci of the central conic. It predicts that, among the stated 3-Poncelet pairs, constant area throughout the Poncelet family occurs exactly for concentric circular pairs.