The Gaussian correlation inequality for disjoint Wishart principal minors

Let $\mathfrak{X}$ be the Wishart random matrix in the setting of the cited theorem, partitioned into diagonal blocks $\mathfrak{X}_{ii}$ of widths $p_1,\ldots,p_d$. Wishart principal-minor Gaussian correlation conjecture. For every $k\in\{2,\ldots,d\}$ and $(t_1,\ldots,t_d)\in(0,\infty)^d$,

$$\mathsf{P}\left(\bigcap_{i=1}^d\{|\mathfrak{X}_{ii}|\leq t_i\}\right)\geq\mathsf{P}\left(\bigcap_{i=1}^{k-1}\{|\mathfrak{X}_{ii}|\leq t_i\}\right)\mathsf{P}\left(\bigcap_{i=k}^d\{|\mathfrak{X}_{ii}|\leq t_i\}\right).$$

The conjecture is motivated by Gaussian correlation inequalities for the associated multivariate Gamma distribution and would provide an alternative route to a lower bound for products of principal-minor functions. The supplied text gives no resolution status.