Proof of Gaussian moment product conjecture

For an $n$-dimensional real-valued centered Gaussian random vector $(X_1,\dots ,X_n)$ with any covariance matrix, the following moment product conjecture is proved in this paper $$\mathbb{E}\prod _{j=1}^n{X}_j^{2m_j}\ge \prod _{j=1}^n\mathbb{E}{X}_j^{2m_j},$$ where $m_j\ge 1,1\le j\le n,$ are any positive integers. Among other important applications, a special case of this conjecture (with $m_j=m,1\le j\le n$) would give an affirmative answer to another open problem: real linear polarization constant. The proof is based on a very elegant and elementary approach in which only one component $X_j$ of the random vector is chosen with varying variance.