Structure of simplex volumetric moments as polynomials in even powers of pi

Prove that for every integer r ≥ 0 and every moment order k > −1, the normalized k-th volumetric moment v_{r+1}^{(k)}(T_{r+1}) of a random (r+1)-simplex T_{r+1} can be expressed as a finite linear combination of even powers of π with rational coefficients; that is, establish the identity v_{r+1}^{(k)}(T_{r+1}) = ∑_{s=0}^{⟂r/2⟂} p_{r s}^{(k)} π^{2 s} for some rationals p_{r s}^{(k)}.

Background

Throughout the paper the authors compute many exact odd volumetric moments for simplices in dimensions 3–6 and observe that the results involve only even powers of π with rational coefficients (and, in higher dimensions, possibly values of polylogarithms).

Motivated by these patterns, they formulate an explicit conjecture describing the general algebraic form of v_{r+1}^{(k)}(T_{r+1}). Proving or refuting this conjecture would clarify the structure of volumetric moments for random simplices across dimensions.