  This paper extends various results related to the Gaussian product inequality
(GPI) conjecture to the setting of disjoint principal minors of Wishart random
matrices. This includes product-type inequalities for matrix-variate analogs of
completely monotone functions and Bernstein functions of Wishart disjoint
principal minors, respectively. In particular, the product-type inequalities
apply to inverse determinant powers. Quantitative versions of the inequalities
are also obtained when there is a mix of positive and negative exponents.
Furthermore, an extended form of the GPI is shown to hold for the eigenvalues
of Wishart random matrices by virtue of their law being multivariate totally
positive of order~2 ($\mathrm{MTP}_2$). A new, unexplored avenue of research is
presented to study the GPI from the point of view of elliptical distributions.
