Pseudodistribution product inequality for semidefinite quadratic forms

Let $A_1,\ldots,A_d$ be the positive semidefinite matrices in the paper's optimization problem, let $\mathcal{S}_k$ denote the family of $k$-element subsets of $\{1,\ldots,d\}$, and let $\mu(x)$ be the normalized weight

$$\mu(x)=\frac{\langle v,x\rangle^{2(k-1)}}{\mathbb{E}_x[\langle v,x\rangle^{2(k-1)}]}.$$

Pseudodistribution product inequality.

$$\left(\prod_{i=1}^d\mathbb{E}_x\left[\mu(x)\langle x,A_i x\rangle\right]\right)^{\binom{d-1}{k-1}} \geq \prod_{I\in\mathcal{S}_k}\mathbb{E}_x\left[\prod_{i\in I}\langle x,A_i x\rangle\right].$$

If true, this inequality would imply that $\textsc{OptSOS}_k(\mathcal{A})$ has approximation factor $e^{-C(n,k)}$. The supplied span does not establish resolution, and the notation for the distributions, expectation functional, and matrix assumptions is only partially present in the excerpt.