Polar-pentagon centroid-locus conjecture

Let $\cal{P}$ be a parabola and consider the family of polar polygons associated with parabola-inscribed Poncelet pentagons. Polar-pentagon centroid-locus conjecture. The loci of the vertex and area vertices are lines perpendicular to the directrix, whereas the locus of the perimeter centroid is an algebraic curve of degree at least four. The claim is presented as an experimentally observed distinction between the vertex or area loci and the perimeter-centroid locus.