The general sparse sum-of-squares certificate conjecture for Vizing's conjecture

Let $\mathcal{G}$ and $\mathcal{H}$ be the graph objects used in the paper's polynomial model, with vertex sets $V(\mathcal{G})$ and $V(\mathcal{H})$, parameters $n_\mathcal{G}$ and $n_\mathcal{H}$, and $k_\mathcal{G}$ and $k_\mathcal{H}$. Let $x_{gh}$ be the associated polynomial variables, let $f^{*}$ be the target polynomial, and let $j$-sos certificate mean a sum-of-squares certificate of the indicated degree. General sparse certificate conjecture. If $k_\mathcal{G}=n_\mathcal{G}$ and $k_\mathcal{H}=n_\mathcal{H}-j$ for $j\geq 3$, then

$$s_g=\sum_{q=0}^{j}\alpha_q\left(\sum_{\substack{S\subseteq V(\mathcal{H})\\|S|=q}}\prod_{h\in S}x_{gh}\right)\quad\text{for }g\in V(\mathcal{G}),$$

where the $\alpha_q$ solve a certain system of polynomial equations, gives a $j$-sos certificate of $f^{*}$. The claim is motivated by certificates established for the preceding cases, but the paper presents it as an expected generalization rather than a proved result.