Results related to the Gaussian product inequality conjecture for mixed-sign exponents in arbitrary dimension

This note establishes that the opposite Gaussian product inequality (GPI) of the type proved by Russell & Sun (2022a) in two dimensions, and partially extended to higher dimensions by Zhou et al. (2024), continues to hold for an arbitrary mix of positive and negative exponents. A general quantitative lower bound is also obtained conditionally on the GPI conjecture being true.