Zero-rotation locus conjecture for the circumcenter map of regular polygons

Let $\mathcal{P}$ be a regular $n$-gon, with $n\geq 3$, and let $\mathcal{P}'=\mathcal{C}_M^n(\mathcal{P})$. Write $\alpha$ for the angle of rotation in the similarity taking $\mathcal{P}$ to $\mathcal{P}'$. Zero-rotation locus conjecture. The locus of points $M$ such that $\alpha=0$ is the union of $n$ lines through the centroid of $\mathcal{P}$, with directions $k\pi/n$ for $k=0,\ldots,n-1$. This claim is presented as a conjecture following computational and visual evidence, and no proof or resolution is supplied.