White's three equivalence-relation conjectures for matroid bases

Let $M$ be a matroid, and let $TE(i)$ be the class of matroids for which every pair of compatible sequences of bases is equivalent under White's relation $\sim_i$, for $i=1,2,3$. Here compatibility means that the two sequences have equal unions as multisets; $\sim_1$ is generated by symmetric exchanges, $\sim_2$ additionally permits permutations of bases, and $\sim_3$ is generated by multiple symmetric exchanges.

White's $TE$ conjecture. All matroids satisfy each of the three properties:

$$TE(1)=TE(2)=TE(3)=\text{the class of all matroids}.$$

The source explicitly describes this as White's original conjecture and states that it is an open question. The $TE(2)$ assertion is the toric-ideal conjecture above, while the three equalities are retained here because the source states them as a single conjecture.