General parabola centroid-locus conjecture for Poncelet polygons

Let $\cal{P}$ be a parabola and $\cal{C}$ a focus-centered circle, and let $\cal{R}$ be a Poncelet family of $N$-gons inscribed in $\cal{P}$ and circumscribed about $\cal{C}$. General parabola centroid-locus conjecture. For arbitrary $N$, the loci over $\cal{R}$ of the vertex, perimeter, and area centroids are parabolas coaxial with $\cal{P}$. The conjecture is motivated by evidence for $N=3,4,5,6$ and remains unproved in the supplied text.