Gaussian product inequality conjecture

Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. Then, for every $\alpha_1,\ldots,\alpha_n > 0$, $\mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right].$ Moreover, if $\mathsf{Var}(X_i) > 0$ for every $i$, then equality holds if and only if $X_1,\ldots,X_n$ are independent.