Combinatorial positivity conjecture for the Gaussian product inequality

Let $m_2\geq 2$ and $m_3\in\mathbb{N}$. Let $c^{m_2,m_3}_p$ denote the coefficients introduced in the paper's combinatorial formulation of the three-dimensional Gaussian product inequality.

Combinatorial positivity conjecture.

$$c^{m_2,m_3}_p>0\qquad\text{for }3\leq p\leq 2m_2-1.$$

This coefficient-positivity claim is intended to imply the corresponding three-dimensional Gaussian product inequality. The source verifies related coefficient properties and small cases, but the stated range remains open there.