Unimodular flag triangulation conjecture for polymatroid base polytopes

Let $P$ be the base polytope of a polymatroid. A triangulation of $P$ is unimodular if all its simplices are unimodular with respect to the relevant lattice, and it is flag if every minimal nonface has two vertices. Polymatroid triangulation conjecture. Every polymatroid base polytope has a unimodular flag triangulation. The conjecture is motivated by the relationship between flag triangulations, quadratic Gröbner bases and Koszul algebras; the source presents it as supported by experiments but gives no resolution.