Characterization of triangle centers with elliptic loci in concentric Poncelet pairs

Let $\mathcal{X}$ be a triangle center, and consider its locus as the triangle ranges over Poncelet 3-periodics in a concentric pair of non-axis-aligned ellipses. Elliptic-locus characterization conjecture. If the locus of $\mathcal{X}$ is an ellipse for every such pair, then $\mathcal{X}$ is a fixed linear combination of $X_2$ and $X_3$. The statement is marked as subsequently proved in the source's status evidence, so it is recorded as solved; the cited proof attribution is not bibliographically expanded in the supplied material.