Ehrhart positivity conjecture for matroids with series-parallel subdivisions

Let $M$ be a connected matroid, and let $P_M$ be its base polytope. Say that $P_M$ admits a series-parallel subdivision if it can be subdivided into base polytopes of series-parallel matroids. A matroid is Ehrhart positive when its base polytope has an Ehrhart polynomial with nonnegative coefficients. Series-parallel subdivision conjecture. If $P_M$ admits a series-parallel subdivision, then $M$ is Ehrhart positive. This conjecture would imply the conjectured Ehrhart positivity of positroids and, because transversal matroids admit such subdivisions, would also imply positivity for transversal and cotransversal matroids. It is open in the supplied text.