Conjecture on invariant sums of squared side areas for cyclic polygons

Let $\mathcal P$ be a family of cyclic $n$-gons circumscribed about a central conic. For each polygon in the family, construct a square on every side and sum the areas of these squares. The circumcenter of the triangle is the center of $\mathcal D$ or one of its foci.

Invariant side-square area conjecture. The sum of the areas of the constructed squares remains invariant throughout $\mathcal P$ if and only if the circumcenter of the triangle coincides either with the center of $\mathcal D$ or with one of its foci.

This conjecture proposes an extension of the stated triangle theorem to cyclic $n$-gons circumscribed about a central conic. Its status is unresolved in the supplied text.