The strong Gaussian product conjecture for even exponents

Let $n\in\mathbb{N}$ and let $p_1,\dots,p_n\in 2\mathbb{N}$. For a real centered Gaussian vector $(X_1,\dots,X_n)$, the statement $\mathbf{GPI}_n(p_1,\dots,p_n)$ means

$$\mathbb{E}\left[\prod_{i=1}^n |X_i|^{p_i}\right]\geq\prod_{i=1}^n\mathbb{E}\left[|X_i|^{p_i}\right],$$

with equality if and only if $X_1,\dots,X_n$ are independent. Strong Gaussian product conjecture. For all $n\in\mathbb{N}$ and all $p_1,\dots,p_n\in 2\mathbb{N}$, $\mathbf{GPI}_n(p_1,\dots,p_n)$ holds. The result in the paper proves the case $n=3$; the conjecture remains unresolved in general.