Conjecture on optimal quantizers for the uniform distribution on a disc

Let the unit disc be equipped with the uniform probability distribution. For an optimal set of $n$-means, consider the associated Voronoi regions. Conjecture on optimal quantizers for the disc. (i) For $n=2$, the Voronoi regions partition the disc into two regions bounded by semicircles. (ii) For $n=3$, they partition the disc into three sectors, each subtending a central angle of $\frac{2\pi}{3}$ radians. (iii) For $n=4,5,6$, the Voronoi regions either form a regular $n$-gon centered at the center of the disc, or one point lies at the center of the disc and the other $n-1$ points form a regular $(n-1)$-gon centered at the center of the disc. These configurations are motivated by the maximum geometric and distributional symmetry of the uniform disc; the claims concern the unresolved structure of optimal quantizers for these values of $n$.