Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for joint distribution of traces Wishart matrices. In particular, several GPI-related from Wei [32] and Liu et al. [15] are extended in two ways: by replacing power functions with more general classes functions, usual multivariate gamma distributional assumptions trace-Wishart assumption. These findings suggest that a Kronecker form GPI holds diagonal blocks any distribution.