Three-dimensional Gaussian product inequality conjecture

Let $m_1,m_2,m_3\in\mathbb{N}$, and let $(X_1,X_2,X_3)$ be a centered Gaussian random vector. Three-dimensional Gaussian product inequality conjecture.

$$E[X_1^{2m_1}X_2^{2m_2}X_3^{2m_3}]\geq E[X_1^{2m_1}]E[X_2^{2m_2}]E[X_3^{2m_3}].$$

The equality sign is asserted to hold if and only if $X_1,X_2,X_3$ are independent. The paper describes the validity of this three-dimensional inequality as still open in the general case, while proving a special case with one exponent equal to one.