The Gaussian Product Conjecture

Let $d\geq 2$, let $(G_1,\ldots,G_d)$ be a $d$-dimensional real-valued centered Gaussian vector, and let $m\geq 1$ be an integer. Gaussian Product Conjecture. One has

$$\mathbb{E}[G_1^{2m}\cdots G_d^{2m}]\geq\prod_{i=1}^d\mathbb{E}[G_i^{2m}].$$

This is an open problem and a natural real counterpart to the corresponding complex Gaussian moment inequality; its solution would imply the Real Polarization Problem.