The one-dimensionality conjecture for analytic centers of simplicial polytopes

Let a center assign a point to every simplicial polytope in ${\mathbb R}^n$. Assume that the center depends analytically on the polytope, commutes with dilations, and satisfies the Archimedes Lemma with weights equal to the respective volumes. One-dimensionality conjecture. The space of such centers is 1-dimensional: every such center is an affine combination of the center of mass and the circumcenter of mass. The claim proposes a classification of centers satisfying these natural geometric and analytic conditions; the supplied text gives no resolution, so its status remains open.