TITLE: An inequality for expected value of normally distributed variables
QUESTION [13 upvotes]: Question. Let $X_1,\dots,X_n$ be random variables with normal distribution. Is it true that
  $$\mathbb E \prod_{i=1}^nX_i^{2k}\ge\prod_{i=1}^n\mathbb E X_i^{2k}$$for any $k\in\mathbb N$?

(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for a solution is "zywa kura", see pages 37,38 in the Volume 1 of the Lviv Scottish Book. To get the prize, write to the e-mail: hsc@pwr.edu.pl).

REPLY [8 votes]: The problem posed above is a semi-well-known open problem that I believe is equivalent to the real polarization conjecture. A more general version of the question posed above is offered as Conjecture 4 in this paper of Wenbo Li, who considers arbitrary positive powers of the type $|X_i|^{p_i}$.
For the case $k=1$, a proof is known. Have a look at Theorem 2.1 in the linked paper for a proof of $k=1$ case (thanks to Iosif for alerting me that my answer was only about the $k=1$ case). In particular, the variables have to be jointly Gaussian with mean zero. Moreover, equality holds if and only if they are independent or at least one of them is a.s. zero.