The Gaussian product inequality conjecture for multinomial covariances

In this paper, we find an equivalent combinatorial condition only involving finite sums (which is appealing from a numerical point of view) under which the centered Gaussian random vector with multinomial covariance, $(X_1,X_2,\dots,X_d) \sim \mathcal{N}_d(\boldsymbol{0}_d, \mathrm{diag}(\boldsymbol{p}) - \boldsymbol{p} \boldsymbol{p}^{\top})$, satisfies the Gaussian product inequality (GPI), namely $$\mathbb{E}\left[\prod_{i=1}^d X_i^{2m}\right] \geq \prod_{i=1}^d \mathbb{E}\left[X_i^{2m}\right], \quad m\in \mathbb{N}.$$ These covariance matrices are relevant since their off-diagonal elements are negative, which is the hardest case to cover for the GPI conjecture, as mentioned by Russell & Sun (2022). Numerical computations provide evidence for the validity of the combinatorial condition.