The rank-three transversal matroid half-plane-property conjecture

Let $M$ be a transversal matroid of rank $3$. Its basis generating polynomial is the sum of the monomials corresponding to the bases of $M$.

Rank-three transversal matroid conjecture. Every rank-$3$ transversal matroid has the half-plane property.

The conjecture is motivated by numerical experiments and by the fact that the paper proves the half-plane property only for a subclass of transversal matroids, called nice transversal matroids. The authors report no known counterexample, but do not provide a proof.