Polar-parabola centroid-locus conjecture for arbitrary polygon size

Let $\cal{P}$ be a parabola, let $\cal{C}$ be a focus-centered circle, and let $\cal{R}$ be a Poncelet family of $N$-gons inscribed in $\cal{P}$ and circumscribed about $\cal{C}$. Form the polar polygons of $\cal{R}$ with respect to $\cal{P}$. Polar-parabola centroid-locus conjecture. For arbitrary $N$, the loci of their vertex and area centroids are straight lines parallel to the directrix of $\cal{P}$. The claim generalizes the observed behavior for the studied polygon sizes and is not resolved in the supplied text.