Toric ideals of lattice path matroids and polymatroids

White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. We prove a stronger version of this conjecture for lattice path polymatroids by constructing a monomial order under which these sets of quadrics form Gröbner bases. We then introduce a larger class of polymatroids for which an analogous theorem holds. Finally, we obtain the same result for lattice path matroids as a corollary.