The chordality characterization conjecture for homaloidal spanning tree polynomials

Let $G$ be an undirected graph, and let $P_G$ denote the spanning tree generating function corresponding to $G$. A polynomial is homaloidal if its polar map is birational. Chordality conjecture. The polynomial $P_G$ is homaloidal if and only if $G$ is chordal. The paper proves homaloidality when $G$ is chordal and non-homaloidality for cycles $C_n$ with $n\geq 4$; the converse direction for general non-chordal graphs remains open.