Neil White's symmetric-exchange conjecture for matroid toric ideals

Let $M$ be a matroid on the ground set $\{1,2,\ldots,n\}$. Fix a field $k$, let $S_M=k[y_B:B$ a base of $M]$, and let $I_M$ be the kernel of the homomorphism $S_M\to k[x_1,\ldots,x_n]$ sending $y_B$ to $\prod_{i\in B}x_i$. Given bases $B$ and $D$, a double swap is a pair $b\in B$ and $d\in D$ such that $B\cup d-b$ and $D\cup b-d$ are bases. Neil White's conjecture. For any matroid $M$, the toric ideal $I_M$ is generated by the quadratic binomials $y_{B_1}y_{B_2}-y_{D_1}y_{D_2}$ such that the pair of bases $D_1,D_2$ can be obtained from the pair $B_1,B_2$ by a double swap. This conjecture is a proposed algebraic form of White's symmetric-exchange conjecture; the paper proves it for graphic matroids, while the general matroid case remains open.