Kontsevich's polynomial-counting conjecture for graph hypersurface complements

Let $G$ be a finite graph with vertex set $V(G)$) and edge set $E(G)$. For each edge $e$, let $x_e$ be a variable, and define the Kirchhoff polynomial

$$P_G= \sum_T\prod_{e\notin T}x_e,$$

where the sum runs over spanning trees $T$ of $G$. Let $V(P_G)$ be the zero scheme of $P_G$ in $\mathbf A^{E(G)}$, let $Y_G=\mathbf A^{E(G)}\setminus V(P_G)$, and, for a scheme $X$ of finite type over $\mathbf Z$, let $[X]$ denote the function $q\mapsto\#X(\mathbf F_q)$. A scheme is polynomially countable if its point-counting function lies in $\mathbf Z[q]$. Kontsevich's conjecture. For all graphs $G$, $[Y_G]\in\mathbf Z[q]$. Since $[V(P_G)]+[Y_G]=q^{\#E(G)}$, this is equivalent to the conjecture that $V(P_G)$ is polynomially countable. The conjecture arose from expectations that graph-hypersurface periods are multiple zeta values and that the associated zeta functions come from mixed Tate motives; it was subsequently shown in the paper to be false.