Quadratic triangulations for dyadic matroid base polytopes

Let $M$ be a dyadic matroid, meaning a matroid representable over the dyadic partial field. Let $P(M)$ be its matroid base polytope. Dyadic matroid triangulation conjecture. The matroid base polytope $P(M)$ has a quadratic triangulation. The source presents this as a proposed problem motivated by observed connections with representability theory; it also notes that the Fano matroid does not have a quadratic triangulation.