Stembridge's polynomial-counting conjecture for nondegenerate sparse symmetric matrices

Let $G$ be a simple graph with vertex set $V(G)=\{v_1,\ldots,v_n\}$. Let $Z^o_G$ be the scheme of nondegenerate symmetric $n\times n$ matrices $M$ satisfying

$$M_{ij}=0\quad\text{if }i\ne j\text{ and there is no edge from }v_i\text{ to }v_j.$$

Equivalently, for a graph $G^*$ obtained from $G$ by adjoining an apex vertex connected to every vertex, the source identifies $Z^o_G$ with $X_{G^*}$. Stembridge's conjecture. For every simple graph $G$, $[Z^o_G]$ is a polynomial in $q$. Since it would follow from Stanley's reformulation, and that reformulation is equivalent to the refuted Kontsevich conjecture, this conjecture is also refuted.