Perimeter-centroid conic conjecture for polar images of bicentric families

Let $\cal{B}'$ be the conic-inscribed polar image of a generic bicentric family of $N$-gons with respect to its bicentric circumcircle. Perimeter-centroid conic conjecture. Over $\cal{B}'$, the locus of the perimeter centroid is a conic. The claim is supported by experimental evidence; the supplied text gives no proof or resolution.