Pentagon centroid-locus conjecture for parabola-inscribed Poncelet families

Let $\cal{P}$ be a parabola of focal distance $f$, and let $\cal{C}$ be a focus-centered circle of radius $r$ such that they admit a Poncelet family of inscribed pentagons. Pentagon centroid-locus conjecture. Over this family, the loci of the vertex, perimeter, and area centroids are parabolas coaxial with $\cal{P}$. This is an experimentally observed claim for the pentagon family; no proof or resolution is supplied.