An inequality for expected value of normally distributed variables 
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). 
 A: 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.
A: Some partial counterexamples: 
Let us assume that "with normal distribution" means "with joint normal distribution". Even then, in general the inequality will not hold. Indeed, 
\begin{equation}
 E(Z-1)^2(Z+1)^2<E(Z-1)^2\;E(Z+1)^2, 
\end{equation}
where $Z\sim N(0,1)$. 
The inequality may hold assuming also that the $X_i$'s are zero-mean. 
However, the stronger inequality 
\begin{equation}
 E \prod_{i\in S\cup T}X_i^{2k}\ge E \prod_{i\in S}X_i^{2k}\  E \prod_{i\in T}X_i^{2k}
\end{equation}
for finite disjoint sets $S$ and $T$ will not hold 
in general (or even when the set $T$ is a singleton) -- even if the $X_i$'s are zero-mean jointly normal. 
E.g., let $X_1=Z_1$, $X_2=Z_1-Z_2$, $X_3=\frac12 Z_1+Z_2$, where the $Z_i$'s are independent $N(0,1)$. Then 
\begin{equation}
 E X_1^2 X_2^2 X_3^2=\frac{15}4<4\times\frac54=E X_1^2 X_2^2 \ E X_3^2. 
\end{equation}
