Regularity maximizes area-contracting regions for the circumcenter map

Let $\mathcal{Q}$ be a simple $n$-gon, meaning that it has no self-intersections, and consider the regions of points $M$ for which the circumcenter map produces area contraction. Regularity-maximization conjecture. For a fixed $n$, the number of area-contracting regions for a simple $n$-gon $\mathcal{Q}$ is maximal when $\mathcal{Q}$ is regular. The statement is motivated by experiments, and the source provides no proof or resolution.