Wishart Gaussian product inequality conjecture

Let $\mathfrak{X}\sim \mathcal{W}_p(\alpha,\Sigma)$ for given $p\in \mathbb{N}$, $\alpha\in (p-1,\infty)$ and $\Sigma\in \mathcal{S}_{++}^p$. For every $\nu_1,\ldots,\nu_d\in [0,\infty)$, consider the diagonal entries $\mathfrak{X}_{ii}$ and their absolute moments.

Wishart Gaussian product inequality conjecture.

$$\mathsf{E}\!\left(\prod_{i=1}^d |\mathfrak{X}_{ii}|^{\nu_i}\right) \geq \prod_{i=1}^d \mathsf{E}(|\mathfrak{X}_{ii}|^{\nu_i}).$$

This is the Wishart analogue of the Gaussian product inequality and is proposed as a framework for obtaining explicit bounds in statistical testing and for investigating algebraic inequalities and analogues of polarization and correlation conjectures. The paper proves a stronger quantitative version when $d=2$, but the general assertion remains open.