The sums-of-squares conjecture for Gaussian product polynomials

Let $n\ge 3$, let $m_1,\dots,m_n\in\mathbb{N}$, and let $U_1,\dots,U_n$ be independent standard Gaussian random variables. Define the polynomial $H$ on $\mathbb{R}^{n(n-1)/2}$ by

$$\begin{aligned} H(x_{21},x_{31},x_{32},\dots,x_{n1},\dots,x_{n,n-1})={}&E\left[U_1^{2m_1}\prod_{i=2}^{n}\left(\sum_{j=1}^{i-1}x_{ij}U_j+U_i\right)^{2m_i}\right]\\ &-(2m_1-1)!!\prod_{i=2}^{n}E\left[\left(\sum_{j=1}^{i-1}x_{ij}U_j+U_i\right)^{2m_i}\right]. \end{aligned}$$

The sums-of-squares conjecture for Gaussian product polynomials states that $H$ has a sums-of-squares representation. This proposed stronger formulation would imply the corresponding even Gaussian product inequalities. The authors motivate it as a route to proving the GPI and report exact sums-of-squares verifications for numerous cases, but do not establish the assertion for all $n$ and $m_j$; it remains open.