Real-rootedness of three-class rank-two matroids with large parallel classes

Determine which rank-two matroids with exactly three parallel classes, each having size at least four, have real-rooted Ehrhart h*-polynomials.

Background

The paper proves real-rootedness for rank-two matroids with three parallel classes whenever the smallest parallel class has size at most three. It also constructs counterexamples with parallel-class sizes (4,561,600) and (4,a,a+29) for all sufficiently large integers a, showing that real-rootedness can fail when the smallest class has size four.

Consequently, the unresolved cases among rank-two matroids with exactly three parallel classes include configurations in which all three parallel classes have size at least four. The authors leave open the task of determining precisely which such configurations retain real-rooted Ehrhart h*-polynomials.