Quadraticity conjecture for matroid base polytopes

Let $\mathcal{M}$ be a matroid with basis polytope $P_B(\mathcal{M})$, and let $I_{P_B(\mathcal{M})}$ be its toric ideal.

Matroid-base quadraticity conjecture. The toric ideal $I_{P_B(\mathcal{M})}$ is quadratic.

This is presented as a weaker version of White's conjecture. The source does not give resolution evidence for this weaker statement, so its status is recorded as open.