Distinguishing loopless lattice-path matroids

Determine whether loopless lattice-path matroids are distinguished by their chromatic quasisymmetric function.

Background

Lattice-path matroids form a broader class than the nested or Schubert matroids that play a central role in the paper. The question parallels the well-known problem of whether chromatic symmetric functions distinguish trees, but asks it for loopless lattice-path matroids and the matroid chromatic quasisymmetric function.

The problem concerns the injectivity of the invariant on this class: two loopless lattice-path matroids would need to have different chromatic quasisymmetric functions unless they are appropriately identified.