The quadratic symmetric-exchange generation conjecture for matroid toric ideals

Let $M$ be a matroid, with toric ideal $I_M$. For bases $B_1,B_2,D_1,D_2$, call $y_{B_1}y_{B_2}-y_{D_1}y_{D_2}$ a double-swap binomial when the pair $D_1,D_2$ is obtained from $B_1,B_2$ by a double swap. The quadratic symmetric-exchange generation conjecture. For any matroid $M$, the quadratic binomials of $I_M$ are in the ideal generated by the double-swap binomials $y_{B_1}y_{B_2}-y_{D_1}y_{D_2}$. The source identifies this as the second of two open conjectures that together imply White's conjecture.