  This note reports partial results related to the Gaussian product inequality
(GPI) conjecture for the joint distribution of traces of Wishart matrices. In
particular, several GPI-related results from Wei (2014) and Liu et al. (2015)
are extended in two ways: by replacing the power functions with more general
classes of functions, and by replacing the usual Gaussian and multivariate
gamma distributional assumptions by the more general trace-Wishart distribution
assumption. These findings suggest that a Kronecker product form of the GPI
holds for diagonal blocks of any Wishart distribution.
