The Gaussian product inequality for disjoint principal minors of Wishart matrices

Let $p\in\mathbb{N}$, let $\alpha\in(p-1,\infty)$, and let $\Sigma\in\mathcal{S}_{++}^p$. Let $\mathfrak{X}\sim\mathcal{W}_p(\alpha,\Sigma)$ be a $p\times p$ Wishart random matrix, partitioned into blocks $(\mathfrak{X}_{ij})_{1\leq i,j\leq d}$, where block $\mathfrak{X}_{ij}$ has size $p_i\times p_j$ and $p_1+\cdots+p_d=p$. Wishart principal-minor Gaussian product inequality. For every $\nu_1,\ldots,\nu_d\in[0,\infty)$,

$$\mathsf{E}\left(|\mathfrak{X}_{11}|^{\nu_1}\cdots|\mathfrak{X}_{dd}|^{\nu_d}\right)\geq\mathsf{E}\left(|\mathfrak{X}_{11}|^{\nu_1}\right)\cdots\mathsf{E}\left(|\mathfrak{X}_{dd}|^{\nu_d}\right).$$

This is the paper's new conjecture, motivated by results for traces and determinants of disjoint diagonal blocks of Wishart matrices. It is stated as a natural extension to blocks of arbitrary widths; no proof or resolution is supplied.