We present arguments in favor of the inequalities var(Xn2∣X∈Bv(ρ))≤2λnE[Xn2∣X∈Bv(ρ)], where X∼Nv(0,Λ) is a normal vector in v≥1 dimensions, with zero mean and covariance matrix Λ=diag(λ), and Bv(ρ) is a centered v-dimensional Euclidean ball of square radius ρ. Such...