White's conjecture on equivalence of matroid basis sequences

Let $M$ be a matroid, and let

$$\mathcal{X}=(X_1,\dots,X_m),\qquad \mathcal{Y}=(Y_1,\dots,Y_m)$$

be sequences of bases of $M$ of the same length. Two basis sequences are equivalent if one can be obtained from the other by a composition of symmetric exchanges between two sequence positions. They are compatible if, for every ground-set element $s$, it occurs equally many times in the two sequences:

$$\left|\{i:s\in X_i,\ 1\leq i\leq m\}\right|=\left|\{i:s\in Y_i,\ 1\leq i\leq m\}\right|.$$

White's conjecture. The sequences $\mathcal{X}$ and $\mathcal{Y}$ are equivalent if and only if they are compatible. Compatibility is evidently necessary, and the conjecture is connected with toric ideals, Gröbner bases, and the connectivity of basis-pair graphs. It remains open even for sequences of length two.