Three-dimensional Gaussian product inequality with positive integer order moments

The three-dimensional Gaussian product inequality conjecture states that for all positive real numbers p₁, p₂, and p₃, and for all R³-valued centered Gaussian random vectors (X₁,X₂,X₃)^⊤ with Var(X_i)>0, i=1,2,3, the inequality E[|X₁|^p₁|X₂|^p₂|X₃|^p₃]≥E[|X₁|^p₁]E[|X₂|^p₂]E[|X₃|^p₃] holds with equality if and only if X₁,X₂ and X₃ are independent. Recently, Herry, Malicet, and Poly (2024) showed that this conjecture is true when p₁, p₂, and p₃ are even positive integers. We extend this result to any positive integers p₁, p₂, and p₃.